His character could develop a greater sense of humility. If he encounters new and powerful magic or beings in his adventure, he'll realize there's always more to learn in the wizarding world. He may start to see himself as just a small part of a much larger magical universe.
In fanfic, Dumbledore's next great adventure could be a journey through time to correct past mistakes. He might go back to his early days and prevent the rise of a certain dark wizard who influenced his old friend Grindelwald.
His character could be more carefree. In a different AU, he might not have the burden of fighting Voldemort. So he could be more relaxed and focus on things like magical research or teaching without the constant threat of dark forces. For example, he could be more experimental with his teaching methods at Hogwarts.
If Dumbledore traveled to the past and influenced important historical figures, it could change the political and social structure of the wizarding world. Say he convinced a key figure to support a different ideology. This could lead to different laws and norms in the present. It might also change the relationships between different magical communities, like how wizards interact with goblins or centaurs.
Some fanfictions show that Dumbledore gradually reveals more of his trust in Snape to the rest of the Hogwarts staff. This causes Snape to feel a sense of obligation towards Dumbledore. He starts to be more open about his concerns regarding the safety of the school and Harry, especially with Dumbledore. Their relationship becomes more of a partnership, with each relying on the other's unique skills and knowledge.
His relationships would be significantly different. With Ron and Hermione, he might not form the same close bond as in the original story. Since Dumbledore is more of an authority figure, Harry might be more aloof from his peers at first. But he could also attract different friends who are interested in Dumbledore's teachings.
His character could develop by becoming more empathetic. Meeting the next generation, who are more open - minded, might make him realize the harm of his previous bigoted views. He could start to question his Slytherin upbringing and the pure - blood supremacist ideas he was taught.
Hermione's character could develop by becoming more confident in her own instincts. Following Dumbledore, she may start to trust her own judgment more as she observes his actions. For instance, if she disagrees with one of his decisions, she might be more assertive in voicing her opinion, which is a growth from her more deferential nature in the original books.
She becomes more confident in her decision - making. Following Dumbledore's orders, she often has to make quick and important choices. This forces her to trust her instincts more, which in turn makes her more self - assured. For instance, when she has to decide whether to enter a dangerous area to retrieve an important item for Dumbledore, she learns to weigh the risks and benefits on her own and act accordingly.
Well, often there's the theme of power and responsibility. Just like in the main Harry Potter story, being related to Dumbledore means she might have a certain amount of power or influence. She has to learn how to use it responsibly. Also, the theme of friendship can be strong. She'll likely meet other characters and form bonds with them. And mystery is another theme, as there are always questions about Dumbledore's past and she might be involved in uncovering some of those secrets.
His character would become more confident in combat. With Wolverine's powers, he'd feel almost invincible, which could make him a bit cocky at first but then more mature as he learns to handle the power responsibly.