More of a meta-example than anything, but as anyone who went to the midnight releases can attest; for a series that, after the early "Pokemania" phase, largely fell out of the public eye and became known for having a polarized fandom, it's both awesome and heartwarming to see so many people show up for a game that was a part of their (and many other people's) childhoods.
As a meta-example for Christmas 2013, Christmas Wonder Trade◊. Hope the newbie kids enjoy their traded items and Pokémon.
In the beginning of the game, your mother's pet Rhyhorn saying goodbye when you leave on your journey.
Whenever it gives you a ride.
When you pick your starter, it seems happy you chose it and ready to be your partner, just like that.
Some of the reactions of your Pokémon when you play with them in Pokémon-Amie are adorable.
When playing on Pokémon-Amie you can speak or whistle to your Pokémon and they will hear your voice and perk up at the sound of it.
Example, your Honedge will become more cautious in not trying to drain your life energy.
Or your Gourgeist will give up its sadistic ways of cursing and harming others.
Mewtwo's favorite places to be petted are it's hands.
While it can be saddening to see your Pokémon lash out at you for touching them in the wrong spot, sometimes they do it for your own safety.
The Pokédex mentions that anyone who tries to grab Honedge have their life sucked away by its tassel. Honedge doesn't like having its tassel touched, meaning it's trying to protect you. Additionally, Honedge's animation has it holding its sheath with its tassel, allowing it to battle. However, when it's being played with in Pokemon-Amie, its sheath is over its blade. It doesn't want to accidentally cut you.
Espurr doesn't want you to touch its ears because it might lose control of its powers and accidentally kill you.
Speaking of Espurr, it's really cute to see it lighten up and smile/laugh when it's playing with you. Especially since the poor thing's probably lived in fear its whole life.
Roselia does not like it when you touch its flower hands. Considering that there's poison in them, it's likely because it doesn't want you to be poisoned accidentally.
Larvesta dislikes having its horns touched, because of its ability to shoot flames from them. It doesn't want to burn you accidentally.
If you max out your Pokémon's affection, your call out phrases when you're sending them into battle changes, giving them encouragement and suchlike. Additionally, there's a chance that they'll shrug off status changes, such as, when paralyzed, gathering all of its power to instantly break out of the paralysis and, when asleep, shaking itself awake to prevent you from worrying. Also, the Pokémon will sometimes survive an attack with one HP, and the text will read "[Pokémon] toughened it out to show its best side to [Trainer]".
Also, you don't have to worry about an opponent Pokémon's attack hitting at all, as sometimes, according to the text, before the attack can even hit, your Pokémon will read your mind and dodge the attack.
When an opponent is defeated, holding the stylus on the Touch Screen before the Pokémon's defeated animation ends will cause you to afterwards pet your Pokémon on the head as though to say, "Good job! You did well!"
One of the more memorable messages you can get is '[Pokemon] is remembering the first time it met [Trainer]...'
Taken Up to Eleven if you're playing a Nuzlocke run and you're met with a situation where your Pokémon would almost certainly be dead but it survives or avoids an attack like that.
Also consider the fact that in previous gens the games have always acted as though you have strong bonds of love and friendship with your Pokémon and that's why you triumph on your journey, but it was mostly just an Informed Attribute. So actually getting to foster those bonds and seeing the power of love and friendship in action is awesome.
Just as heartwarming is getting to interact with a Pokemon you transferred from a previous game. These Pokemon are probably special to you, and to get the chance to interact with them in this manner is nothing short of beautiful.
As mentioned in the funny entry, Yveltal can be this in Amie. A dark-flying Pokémon - the very incarnation of destruction, capable of killing everything around it so it can revive - is shown to be a quite benevolent and tame creature once you befriend it. It acts like a baby Pokémon when you feed it, pet it or play with it; and it smiles and cries in happiness quite a lot during all that. Dark Is Not Evil at its finest.
Heck, Amie as a whole can reveal a hidden side out of any Pokémon: You know about Bisharp? A literal combination of a Corrupt Samurai and a Mob Boss, looks cool as ice, but you pet it and feed it a Pokepuff and it will let out a hearty laugh.
When you turn on Wonder Trade or connect to the Internet, other people's Pokémon will come by, talk to your Pokémon, and leave gifts. D'awww!
Meeting Korrina's extra Lucario for the first time. She notes how it seems attached to you and when she leaves, it takes a moment to leave you and follow her. It joins you later in the game!
During your parade after defeating the Pokémon League, the mysterious Trainer AZ asks to battle you. He's not as strong as the Elite Four, but after you beat him, he finally lets go of the anger in his heart about what happened 3000 years ago. Suddenly, from the sky floats down a Floette - and AZ collapses to his knees and cries with happiness, having finally been reunited with his friend after so long. Also counts as a Tear Jerker.
The entire credits is this in spades. Someway in the credits, lyrics start showing on the bottom screen. At first, it shows the words in two languages: the language you selected and either Frenchnote for English and Asian languages or Japanesenote for other European languages. Then it starts showing multiple languages. The meaning of these words? Cherish life and enjoy it, giving and gratitude does more than taking ever will, and those things will lead to a beautiful world. Sounds cheesy, and the whole setup is a bit more Disney Animated Canon than Pokémon, but it will make at least somebody cry.
It helps that the game has many examples of this happening. A lot of NPCs will give you items for free, while players have plenty of opportunities to fulfill those words: tipping NPCs, giving away rare Pokémon on Wonder Trade, using O-Powers to help people. The game can really bring the best out of people.
Some people may think that the AZ story is an Ass Pull. Some may reckon that the lyrics in the credits is a bit too cheesy. All will agree, however, that the finale itself is a much needed case of both this and Crowning Moment Of Awesome, that really should have been shown in previous Pokemon games. In most previous games, you saving the region from the Villainous Team and becoming the champion are pretty much ignored. Only since the 5th Generation do you start to be acknowledged as the hero/champion. This game takes the respect/recognition Up to Eleven; Professor Sycamore holds a parade specifically for you, and invites people to it. And people actually attend the parade with the Professor awarding you the Legion d'Honneur, specifically because of your heroism! Finally, you get the respect that you really wanted!
There's a woman in Anistar City who can read your Pokémon's memories of your journey together. Sometimes the results can be a bit silly (such as a Pokémon feeling sorry for its trainer using a bicycle) but it's touching when a Pokémon remembers how proud it felt when it evolved, or the impact the journey through Reflection Cave had on it, with all the mirrors within.
There is also another woman who says that she wishes she still has Pokemon when she has children so she can trade with them. Also doubles as being a bit of an Audience Surrogate for the older demographic.
There is another NPC, a middle-aged worker, who says something like "Kids and old guys can both enjoy battling Pokémon!" This is more of a meta-example, since the fact that Game Freak is acknowledging and encouraging the Periphery Demographic is heartwarming in itself.
When you KO a Kangaskhan, the baby gives his mother a hug as they return to the Poké Ball, like it's saying "It's okay Mom, you did your best, I still love you."
Mega Kangaskhan has a lot of this when you stop to think about it. When any other Pokémon Mega Evolves, it uses its Mega Stone to undergo a temporary transformation that grants it more power. When Kangaskhan Mega Evolves, the mother itself stays completely unchanged; rather than use the power itself, it instead selflessly gives the energy to its child, allowing the baby to Mega Evolve and gain the strength to fight alongside its parent. Combined with the Parental Bond ability and the Mega Evolution animation (where the baby leaps out of the pouch and high-fives the mother), it's touching to think about.
In a similar vein with the other Pokémon-Amie reactions above, Kangaskhan will warn you off with a growl if you try to pet her baby instead of you know, outright killing you. Not only that, but when you feed her Pokepuffs, you need to hold it directly between the baby and the mother so they can both eat it at the same time. Their bond is so strong.
The first time you max Affection and get this or similar in battle:
<Pokémon> looked at <your name> with trusting eyes!
Whenever you send out a Pokémon with three Affection or higher against a Gym Leader or Elite Four member.
<Pokémon> turned its head and nodded in understanding!
At Route 19 before Snowbelle City, there's a bridge you have to cross before reaching Snowbelle City. You then encounter Shauna, Tierno, and Trevor, who battle you back-to-back. Shauna battles you, acting all eager, whom you battled with the first time immediately after obtaining your Starter Pokémon, so it could be considered very heartwarming to look back at that (she even mentions that). Then Tierno and Trevor appear, where Tierno asks you if he wants a Pokémon Battle, which is something he never asked before, but remembered what you did back at Geosenge Town, then he says it's his best way to express his thanks. Then Trevor asks for a battle... but not a "Pokédex Battle" like he has been doing throughout the game. He then worries about being weak or strong, then Shauna comments on his being bold. All three of them Took a Level in Badass and all have shown a sign of maturity.
On a meta level, there is a mechanic called "O-power" for use over wireless. It allows you to boost other players in particular ways, whether it be additional money won in battles, sales at Pokémarts, or even a higher catching rate. But it's not limited to your friends and it's not uncommon for random strangers to give you a boost for no reason.
While some people might give O-Powers because they're being nice, one of the reasons is that giving them to someone else costs less orbs than using it on oneself, and it's common courtesy to send them the same power that they sent.
The Looker sidequest has quite a few cute moments. You find a homeless girl named Emma in an alley while on a case, and Looker decides to let her work at the Looker Bureau and care for her as a parent would. Later, Looker has to leave Kalos without you and Emma, so he buys the Bureau so Emma can continue to live there and keep solving cases in Lumiose. Also doubles as a Tear Jerker.
During the 3rd mission, Looker delivers a very heartfelt speech to the Lumiose gang after you defeat them about how he knows people, like Pokémon, can evolve into better members of society. It spurs the gang into setting aside their stealing/bad ways and promising to do good instead, so they can be friends with Emma.
The way Emma smiles at Mimi after commenting that Looker and Xerosic became father-figures to her is absolutely adorable.
When you get to Dendemille Town, you find out the Mamoswine that usually transports trainers to Anistar City on Route 17 is outside the Frost Cavern instead and thus won't give anyone a ride. Why? Because it senses that something's wrong with the Abomasnow in the deepest parts of the cavern (and it turns out it was right because you find Team Flare trying to harness Abomasnow's energy). And if you talk to Abomasnow after you save it, it will give you an Abomasite as a gift of gratitude. You also find out from the man near the Mamoswine that a few years ago (in game), the Abomasnow you saved had carried an injured Mamoswine to town (thus explaining their friendship and Mamoswine's worry).
If you can look beyond its haunting Pokédex entry, you'll find that Phantump is one of the best examples of both Bittersweet and Earn Your Happy Ending in Pokémon ever. As its entry states, it's the spirit of a departed child who was lost in the forest. It's unknown how long it's been a spirit, but it's likely that it will never be reunited with its former parents, even if you try looking for them. But, it's not the end of everything; it has found a kind trainer who carries it with him/her, treating it with love, petting it, feeding it, and playing with it. The way this little critter smiles and enjoys being with you in Pokémon-Amie, it's easy to interpret it as after a long time of sadness and loneliness, you're bringing happiness to it again. By the end, this critter may have lost everything it once had while it was alive, but it now has the one thing it needed more than anything else in this sad yet beautiful world; a good friend.
Not to mention that its evolved form, Trevenant, despite looking scary, is possibly one of the most benevolent Ghost types in the series according to the Pokédex. For one thing, there is exactly one way to draw its ire—hurt the forest it's living in (and given what follows, it probably has to be willful harm, rather than accidental). Given that it expressly prefers old, abandoned forests, your typical lumberjack is probably safe. On top of that, it's genuinely kind to any Pokémon that uses it as a dwelling place. This is not an easily-riled, hateful-towards-the-living ghost. (It helps that it can't learn Spite or Hex.)
Then Fridge Brilliance kicks in when you realize a possible reason why Trevenant is so protective of the forest—the child it used to be (or at least, the girl one of its matrilineal ancestors used to be) died in it. It's not that the forest killed it; rather, the forest adopted it post-mortem. And getting caught/recruited by a trainer isn't a terribly big deal when that likely means getting to meet its adoptive parent's cousins...
Route 12 has a female Pokémon Breeder that before the battle says that her family is tough. She thinks of her Pokémon as family.
While Wonder Trade can sometimes result in someone greedy giving you Com Mons, other times it can result in something quite rare, like a Pokémon with perfect IVs, Egg Moves, and its Hidden Ability. Some of them may be leftovers from breeding, but you definitely know someone is being nice when the Pokémon has Pokérus, an extremely rare status condition that permanently doubles the EVs a Pokémon receives.
A subtle one, but when challenging the Multi Battle Challenge and winding up against Evelyn paired with one of her sisters, it's very clear that they're all supporting her and trying to help draw her out of her shell, in their own different ways.
After beating Wulfric and getting your final badge, he and the Gym guide will come outside to congratulate and encourage you, giving you a final push for the Pokémon League.
When you first enter Pokemon Village, you see Wulfric with some Pokémon (2 Furfrou, a Fletchling, and an Espurr) and when they see you, they hide. Wulfric explains that the Pokémon in here were from abusive trainers and expresses his disappointment in them, but realizes that you're a good trainer and tells the Pokémon it's okay. The Pokémon come back out and if you talk to Fletchling and one of the Furfrou, they give you an item.
A meta example: When the Poké Bank came out in Japan, a number of otherwise-unavailable Pokémon were offered up for Olympus Mons. Some people offered them, then rare and valuable Pokémon, for Com Mons instead.
In Anistar City, there's an old man who asks you if you'll lend him a Pokémon to take care of, because his wife is dead and he's lonely. If you do, after you beat the Elite Four, you find that he's gone, but he left behind your Pokémon and a note thanking you for making his last days happier. Doubles as a major Tear Jerker.
The fireworks at Parfum Palace. Shauna is beyond happy to have the memory of watching them with you, and your hands are just about touching during the show.
Getting 5 hearts of affection on Pokémon-Amie brings up this sweet message once you exit:
Meta example: There are many online forums with people who put up their breeding leftovers for trade which often have optimized natures, egg moves, abilities and several perfect IVs. Many of these people will trade away these Pokemon to new players or those that don't have much so they have something to start with, often for little to nothing in return.
Another Meta example: Pokemon X and Y were the first Nintendo-licensed games ever to be released on the same day in all regions. Considering how socially-driven the Pokemon series is is, and how much international cooperation and friendship is a theme in these two games specifically, it really was a great choice to set the record with.