After rescuing the Fire Daimyo's daughter, Kurama shows how great he's with kids by comforting her, telling her that she's safe and she will see her daddy soon. She immediately dubs him his new best friend and spends all the travel back to Konoha chatting with him.
Kurama is first reluctant to go and see a six-years-old Naruto, fearing to make him a Replacement Goldfish for his best friend. Then he sees how unhappy and alone the kid is, and decides to Screw Destiny, this time Naruto WILL have the love and support he deserves.
And it's even better when you think about it from Naruto's point of view: he was the unwanted orphan everyone shunned, and now he has a loving uncle who would give him the world if he want it.
When he sees how much Kurama is upset by the idea to leave him behind in Konoha, Naruto decides to stay with his newfound uncle, declaring he can still come back when he will be a kickass ninja so awesome that he will be named Hokage, and his first decree would be that Kurama is free to come back too. D'awww.
Despite going to Suna in hopes of hiding Naruto, Kurama ends up repairing Gaara's seal and taking Gaara with him.
When Kurama says to Gaara that his power should be used to protect what's important to him, the kid answers that Kurama is what's important to him. Then Naruto pipes in, saying that Kurama is Naruto and Gaara's precious person, and the Will of Fire is all about protecting that person when you found them. Cue Sand In My Eyes for Kurama.
Momiji's misguided rescue of Fuji. He may think his sister is annoying but he will rescue her from explotative humans.
Chomei sniping at Kurama while at the same time thanking him for saving Fuu.
Why does Roushi decide to help Kurama? Because he remembers being a lonely kid stuck with a demon in his head and the whole world against him, and he knows that Kurama will give to the young jinchurikis the love and protection every child deserve and need.
Takauji and Anzu helping feed and shelter Kurama and the children. They are barely keeping themselves afloat and yet they help them out of gratitude to the Uzumaki.
Kurama returning the traditional Uzu farewell to Takauji, who's delighted to see a little of Uzu live on.
When Kurama dramatically wonders what he did to deserve to be saddled with four brats, Fuu answers it had to be a really great thing, because the kids are his reward. Kurama doesn't deny it.
When he's talking to Kushina, Orochimaru lets slip that Naruto is with her brother. This tidbit of info is twofold for Kushina because it means she's not a Sole Survivor anymore, and her son is safe with someone who would raise hell and worse to protect him. After hearing that, she's more than determined to escape and find her family.
When Kushina encounters Kurama, she immediately recognizes him as the Kyuubi. However, she chooses not to blow his cover and actively reinforces it. All out of gratitude for taking care of Naruto.
This is further sold by how the two strike up a surprisingly real sibling relationship with each other, including how he is the only person allowed to call her 'Tomato' without any real retaliation.
It's violent and slightly disturbing, but Kurama threatening to feed Sakumo his own spleen if the resurrected shinobi doesn't go and talk to his son.
Naruto is disbelieving when Sakumo reveals he's Kakashi's father. Why? Because Sakumo doesn't act like Kurama. Kurama may protest all what he wants, he truly is a Parental Substitute now.
Kakashi quietly confesses that now he understands why his father did what he did... and how proud he is for having such a legacy to live up to.
Rasa finally does right by Gaara when he decides to renounce his parental claim over the boy, entrusting his guardianship to Kurama.