Well, it could be that Dumbledore has some sort of hidden device or spell that he activates when kidnapped. For example, he could have a charmed object that sends a signal to his allies, like the Order of the Phoenix. His allies then come to his rescue and he manages to escape during the chaos. Also, Dumbledore is extremely intelligent. He might outwit Grindelwald by engaging him in a complex conversation full of riddles and misdirections, which gives him an opportunity to break free.
Dumbledore, as Harry's guardian in the fanfiction, would probably use his influence in the wizarding world. He could make sure that Harry is taught by the best teachers at Hogwarts. He'd also be there to offer advice and guidance, especially when Harry faces challenges like dealing with Voldemort or other dark forces. For example, he might teach Harry some advanced defensive magic privately to better protect himself.
He becomes more nurturing. Dumbledore, who was mainly seen as a wise leader, now has to be a caregiver. He might show more patience and tenderness as he takes on the role of a father to Harry.
The relationship would become very strained. Harry's rejection would likely make Dumbledore re - evaluate his actions and how he has interacted with Harry. Dumbledore might be shocked at first, since he probably thought Harry was on the same page as him most of the time.
In many such fanfictions, Dumbledore might comfort Snape by reminding him of his good deeds. For example, he could say that Snape's role as a double agent is crucial for the greater good. Dumbledore might also offer a sympathetic ear, listening to Snape's inner turmoil about his past actions and his love for Lily. By validating Snape's feelings and his importance in the fight against Voldemort, Dumbledore provides comfort.
Typically, it portrays Dumbledore as a manipulator. He's seen as using Snape's feelings for Lily and Harry's role in the fight against Voldemort for his own purposes, rather than having pure intentions.
As the story progresses, Dumbledore might try to make amends. He could start revealing some of his secrets to Harry in an attempt to regain his trust. Harry, on the other hand, would be hesitant at first but gradually start to listen and understand Dumbledore's reasons.
In some fanfics, Harry might hate Dumbledore because he feels Dumbledore has been using him all along as a tool to defeat Voldemort. Dumbledore has been making decisions for Harry without fully considering Harry's feelings and desires. For example, he didn't tell Harry about the prophecy until it was almost too late, and Harry might see this as a form of manipulation.
Dumbledore can hurt Snape in fanfiction by taking advantage of his love for Lily. He might use this love as a tool to manipulate Snape into doing dangerous tasks. For example, making Snape spy on Voldemort, knowing full well that Snape's life is at risk every time he does so.
In my opinion, Dumbledore would underestimate Harry Potter fanfiction. Fanfiction is a realm where fans take the original story and run with it in all kinds of directions. There are fanfictions that change the personalities of characters, create new storylines that deviate greatly from the original. Dumbledore, who is bound by the original narrative and his role in it, may not understand the significance of this new wave of creativity that fanfiction represents. He may see it as a distortion of the world he knows rather than a celebration and expansion of it.