Like a Dragon / Heartwarming

Games with their own pages:

General

Leading up to the American release of Yakuza 6, Sega released a series of live-action shorts called "Stories of the Dragon" featuring people of Kamurocho and elsewhere recounting how Kiryu touched their lives, including a female cabaret club owner whose business Kiryu saved, an aspiring dancer whom Kiryu saved from thugs along with her stepfather, and an old man Kiryu saved from "homeless hunters".
- What makes these promos better is that they're all real substories. "The Bouncer" and "Prodigal Son" from Yakuza Kiwami and "Another Haruka" from Yakuza 5.

Yakuza

The ending, despite how bittersweet it is to make it that far. The fact that Kazuma quickly resigned his role of Chairman, the top position of the Tojo Clan, just so he can have a normal life with Haruka.
After rescuing Haruka in Stardust, Kiryu crouches to her level and finally tells her her mother is dead, breaking into rare tears over having failed her. Despite getting this terrible news right after being kidnapped and even shot, the nine year old's first instinct is to reach forward to touch Kiryu's face and try and comfort him.

Yakuza Kiwami

Kiryu's reunion with Hideki, Mika, Takuma, and Pocket Circuit Fighter. It's touching to look at how he's still fondly remembered by the people he befriended through shared love of an innocent pastime, even after 17 years since they've last seen or heard from him.
Majima getting into Pocket Circuit, of all things. Unlike most of his other interests, which boil down to, "How can I use this to get one over on Kiryu?", he seems genuinely enthralled by it.

Yakuza 3

A great deal many. 3, specifically, has loads of them between Kazuma and his kids. These moments can also overlap with someone getting the living shit beat out of them by an angry Kazuma protecting his kids. DO NOT fuck with Kazuma's kids. EVER.
Some of the moments with the kids include: teaching Shiro to stand up to his bully (the teacher's child), making Izumi feel better about not having a mom or dad by showing her she has a real family (Izumi ran away before dinner. Kazuma told the kids to not worry about waiting for them and that they could begin eating. When they got back to the orphanage, they all waited on Izumi to return before they began eating) and trying to make Mitsuo look cooler so he could impress Riona (another girl at the orphanage).
Saki's drawing of Nakahara and her decision to stay with him.
Mikio shielding Mame's doghouse and taking a sledgehammer to the back of the head to try and protect it.
After the Tear Jerker that is Mine's Redemption Equals Death sacrifice, Daigo asks if he was the traitor and Kiryu reassures him that Mine was not the traitor, because he would never betray Daigo.

Yakuza 4

The reunion between Taiga Saejima and his old boss, Sasai. The former yakuza boss had become homeless, senile, and unresponsive in the years since Saejima was sent to death row, but upon seeing his old charge, something within Sasai awoke...
"...Sa...Sa...Sa...e...ji...ma..."
Goh Hamazaki of all people gets one in the fourth game. In the third game he reacts with scorn to Kazuma's speech about redemption, stabbing him in the gut before asking 'where his redemption is now'. In the fourth game he is obviously touched by Saejima's concern and declarations of brotherhood during their jailbreak and ends up mortally wounded saving him, after he could have quite easily left Saejima behind and got away scot free. Perhaps having a Heel Realisation, he then tells Saejima to find Kazuma when he gets to the mainland and that 'he's a good guy' before taking the sadistic guard with him and allowing Saejima to escape. What really drives it home is Kazuma's reaction when he hears the name. Haruka states that she cannot forgive Hamazaki for what he did, which leads Kazuma to state that if Hamazaki came to him today and asked for forgiveness, he would give it without question. Knowing that as nasty a guy as Hamazaki finally managed to earn the qualities that Kazuma tried to inspire just warms the heart.
Akiyama's relationship with his secretary Hana. Akiyama clearly cares deeply for her and finds her to be beautiful no matter how she looks on the outside, whether she's overweight or thin.

Yakuza: Dead Souls

Akiyama goes to collects a debt from a father who borrowed money to send his daughter that he abandoned to college and his test was giving the money to the daughter himself. However, it turns out he was turned into a zombie. But he leaves the money he owes Akiyama in an envelope with a letter thanking him for his help.
In the ending, Majima leads the reconstruction of Kamurocho. The little girl he met before returns and offers him a homemade bento, which Majima gracefully accepts.
Also in the ending, Ryuji takes over the takoyaki stand, and receives a surprise visit from Kiryu and Haruka. Just goes to show What Could Have Been, at least without a Zombie Apocalypse.

Heartwarming / Like a Dragon