This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Pokémon X and Y
AZ's story. His beloved wife, a Floette, went to fight in a war and was killed. In his grief, he made a machine to bring her back to life. He succeeded, but furious at the world that had killed her, he turned the machine that had resurrected her into a weapon and killed thousands, ending the war. However, Floette, horrified at what he had done, ran away from him. Furthermore, the fallout from the weapon had given the man immortality. So by the end of it, he's stuck alone in a world where he can't die, with nothing but the ruins of a war around him, and being around the people and Pokémon that somehow weren't killed by his weapon or the war. The music in the scene drives it in further.
From AZ's Backstory after his wife went to war: "He was given a tiny box." A rare example of Never Say "Die" actually making a moment even more powerful.
Pokémon Generations' twists the knife with this one by showing that After receiving Floette's coffin, all he can do is grab it, collapse on top of it, and sob hysterically, which is made worse since an interview with Junichi Masuda revealed that the Floette who became his wife was a gift from his mother.
AZ reuniting with Floette at the end. He was Walking the Earth for 3,000 years in repentance for creating the Ultimate Weapon, and after battling the player she finally came back to him... The way AZ shakes is what really sells the scene.
AZ's body language are what really sell the scene. Notice how, once Floette appears, he is trembling, on his knees, visibly shaking at the sight of his wife. Coupled with his shocked facial expression, it is one of the most heartwarming moments in the game, especially since prior to this AZ hardly showed any emotion, let alone the feeble sorrow he displays in this scene.
There have been at least two very large wars in Kalos that ended horrifically, with thousands being killed to end it, the first by AZ's weapon and the second by Yveltal. Kalos may have the bloodiest history of any region we've seen thus far.
The stones on Menhir Trail. They're graves of Pokémon who died in the war.
It's worse than that. They are actually the graves of the Pokémon that were strapped to the stones 3000 ago and lost their souls in order to power the weapon and end the war. Their corpses have just petrified over time and merged with the stones. We even get to see the corpses in AZ's story, which is more than enough to make one cry.
The ending of the Looker subplot in X and Y. Emma only lived with Looker for a few days and already has to say goodbye.
In the post-game, you end up getting hired by Looker. The first two chapters are mostly humorous but comes the third one and you learn that his partner Pokemon apparently died on a mission. The Croagunk you witnessed snatching the Magma Stone away from Charon in Platinum was likely the one being referred to.
It's actually implied to be a few weeks or so, since apparently Looker taught her how to read; that had to have taken a little while. That just makes the end of the sidequest even more depressing, though. They spent enough time together that Emma legitimately saw Looker as a father figure, and one of her last few lines in the ending is wondering if that's what having a dad feels like.
(Which still feels weird regardless since you can finish the entire post-game within a few hours, and since it runs on real-time...)
Heck, the aforementioned parade in Lumoise City where AZ is reunited with his Floette is stated to take place a few days after you beat the Champion... What?
In the front of Lumiose Station, there are two Skiddo sleeping next to each other, which is rather cute. Talk to the old lady in the other side of the street, and she tells you the two of them were abandoned and are waiting there every day for their old Trainer to return. You really want to feel sorry for them and think of all the Pokémon you have released into the wild.
This could be a reference to Hachiko, who also waited for its master at a train station. If this is the case, then the trainer did not abandon them but passed away before being able to pick them up. Either way, it's a Tear Jerker.
Making a return from the Unova games is a more meta one. In one of the houses south of the Pokemon Center in Courmarine City lies someone who looks like an Ace Trainer, and he offers to show you music skills. What song does he play? The poignant "Emotion" that played in multiple key scenes of the Unova Games, which were tear jerkers in their own right.
In a house in Anistar City, there's an old man who is lonely because his wife recently died and he asks you to lend him a Pokemon to keep him company. That's sad enough, right? Well, the player can lend him a Pokémon Level 5 or lower. After the player returns to the home in the post-game, the man is gone and there's a Pokeball on the floor. He's happy that the Pokémon kept him company in his final moments, and thanks the player, and regrets that he will no longer be around to take care of it. The player takes the Pokémon back and also receives a Comet Shard as a gift.
Snowbelle City's Theme. Aside from taking place in a snowy city, it really gives that "near the end" feeling and overall can really be saddening to hear. The anime rendition has been used in a couple sad scenes.
Also the song "Friends Forever", which is heard right before you reach Snowbelle City after you defeat all of your friends in battle and Shauna gives you the Waterfall HM as a final symbol of your friendship.
When you meet Lysandre in Team Flare's hidden base, he reveals his plan to wipe out most of the people on the planet. Calem/Serena asks "What about Pokémon?" Lysandre responds with no words and we can see tears falling from his eyes. He says that Pokémon are wonderful creatures, but with people and Pokémon together, there will always be problems. So there will be no room for Pokémon in his new world order. Meaning he is going to kill all the Pokémon by draining their life force, regretting doing so, but feeling it is the only option.
Despite him being a raving lunatic, it's hard not to feel a little bad for Lysandre when you beat him for the last time; he drops to his knees and lets out an inarticulate scream of despair.
After you've foiled Lysandre's plan, Sycamore confronts you in Couriway Town and thanks you for defeating Lysandre and pins some of the blame for Lysandre's insanity on himself for acknowledging that Lysandre wasn't completely wrong and for not discussing the subject with him.
Then when you arrive in Snowbelle City, an NPC says he is Lysandre's friend and that he used to be a decent person, helping out others, but he snapped when he became fed up with people's stupidity and realizing even he has limitations.
Pokémon-Amie allows you to build your friendship with your Pokémon far more than before in the series, even reflecting this with altered battle text. It's usually sweet, but it can get upsetting when your beloved Partner gets badly beaten up during a battle, and the text says that "It's in a pinch" and "It looks like it's about to cry" as the Low Health Alert jingle beeps at you briefly.
Some of the limitations in Pokémon-Amie are downright tragic in-and-of-themselves. Some Pokémon, like Slugma, cannot be successfully petted without you hurting yourself. Others, like Shedinja, cannot be fed Poké Puffs.
Some of the fainting animations can be this, especially on your favorite Pokémon.
Fainting wild Kangaskhan, as the baby hugs its falling mother.
Even more so if you're doing a Nuzlocke (or, god help you, Wedlocke) run. Or if you've developed such a strong bond with that Pokémon with Pokémon-Amie.
A female Veteran on Route 21. In the process of solving a stone block puzzle, you'll have to walk past a female Veteran, who will challenge you to a battle. Why is she standing where she's standing? She's waiting for a former lover, who swore to her that they would return, and their meeting was to be at that very spot. She only uses one Pokémon (a Tyrantrum). How long has she been standing there? She can't remember. The saddest part of the whole thing is that she has fully realized that her lover will probably never show up (she says that after you beat her), but just can't make herself leave. Just. Ouch.
It sure sucks to be Lysandre's (or Team Flare's) Pokémon, having to follow his command until his goal succeeds. They're either Unwitting Instigator of Doom, Too Dumb to Live (depending on whether they knew the plan), or their Pokéballs are so powerful, they have to follow orders regardless.
Or maybe they share a very strong bond with Lysandre; a bond so strong that they're completely loyal to him to the very end, and are perfectly willing to sacrifice their very lives to let his dream of a beautiful world come true.
It's definitely a very strong bond, considering that Lysandre can cause his Gyarados to Mega Evolve in the 3rd and final battle against him. Said Evolution requires trust and a strong bond between the Pokémon and its Trainer.
Entering the Hall of Fame. In previous games, you were just posed alongside your Pokémon. This time? They gather into a circle facing inward with YOU as the center. It really hammers home that this was a journey both you AND your Pokémon took together.
When you bring a Pokémon that are transferred from Pokémon Black and White or Pokémon Black 2 and White 2 via the Pokémon Bank to the Memory Girl in Anistar City, she tells you the following: "This Pokémon seems to have a good memory, but it doesn't seem to be able to remember..." Essentially, transferring your starter that travelled with you in the beginning of your Unova adventure to Kalos causes it to completely forget about it all, even its old trainer.
It's arguably worse if it's a Pokémon you've transferred up from Pokémon Ruby and Sapphire, or any otherolder game. That Pokémon's been travelling from region to region for years, meeting numerous kind Trainers and amazing rare Pokémon along the way, only for all of it to be forgotten.
It gets really tragic when you send over any Pokémon with N's Trainer ID. If you thought them forgetting the Player was bad enough, it's absolutely heartbreaking that they apparently lost all memories regarding NHarmonia as well. It becomes bittersweet if they hailed from theOrre region as a Shadow Pokémon, as they will forget Wes and Rui/Michael, but also their time as a Shadow, which isn't a bad thing.
On the other hand, they very well might have been unable to remember anything before, and coming to Kalos finally let them remember.
Spare a thought for all those no-longer-canon Generation I and II Pokémon, who not only cannot be transferred forward, but since the internal battery on those games is dying, so are the Pokémon.
It is also possible for the Memory Girl to tell you that your Pokémon is angry with you, if you used an Everstone on it.
Countess Edith in the Battle Chateau will brag about her Pokémon being from her deceased husband before battling you. If you beat her, she'll mournfully state that she just isn't strong enough without him, and talking her again will cause her to reminisce about how he used to look. Well done, you've just trampled on a widow's self-esteem!
Edith (before battle): Now I want you to listen to me...The Pokemon I am about to use were left to me by my dearly departed husband.
Edith (after being defeated): Forgive me darling...I am still so weak without you...
Edith (after battle): My darling husband...Yes, he was a Swimmer (male) with the body of an athlete...
The lyrics that appear on the bottom screen during the end credits. "In our brief lives, we've managed to meet. Treasure this gift. This precious time that we have."
One Preschooler in the Battle Maison says, after you beat him, "I won't cry...I won't. Someday, I will win."
One of the quotes that will appear if your Pokemon has bonded with you enough is "<Pokémon> is remembering the first time it met <Trainer>...". This is especially saddening and heartwarming when you're going through the last Team Flare sequence in the main story, after Lysandre's final Holo Caster announcement.
Let's spare a thought for Korrina for a moment. Note that she has two Lucario, which can't be found in the wild, implying that she caught two Riolus and went from there. How do Riolu evolve, you ask? Through happiness, meaning their bond with the trainer. Then the Lucario that takes a shine to you essentially decides that YOU, a random trainer who's still a novice, is the one who it wants to stay with and fight beside. It probably wasn't a snub towards Korrina, and it's certainly not the player's fault, but that's still a huge, undeserved slap in the face.
Korrina herself actually makes a comment about how she's surprised that her Lucario chose the player over her, but she quickly brushes it aside for the battle. It really makes you wonder if she's really okay with letting her Lucario go with you, or she's just doing it so you won't feel bad.
Just a very minor one: there's a girl in Laverre City who gives you a Gengarite. She mentions that some Pokemon evolve through Link Trades, but since she's all alone, she has no one to Link Trade with. With her mention of Haunter, it's implied she owns one, but can't evolve it into Gengar because there's no one who'll trade with her. It's enough to make you take pity on her, but sadly, the game doesn't offer the chance to perform a Link Trade with her.