Maybe Dumbledore believes that Harry has some hidden power that is being corrupted by the environment at Hogwarts. So, he kidnaps Harry to a secluded place to train him and help him control this power without any distractions or negative influences from other students or even some of the teachers.
One possible motive could be jealousy. Grindelwald might be jealous of Dumbledore's power and popularity in the wizarding world. By kidnapping him, he wants to prove that he is the more powerful one.
Perhaps it's about love in a twisted way. Sebastian might be obsessed with Clary and think that by kidnapping her, he can make her his. In some fanfictions, characters with a warped sense of love do extreme things like this. He could also be trying to use Clary as a pawn in a larger scheme against the Shadowhunters. By having her in his control, he can manipulate the situation to his advantage.
One possible motive could be misinformation. Maybe Bucky was misled to believe the OC was a threat to someone he cares about. So, he kidnapped the OC to prevent them from causing harm.
One possible motive could be protection. Dumbledore might tie Harry to the bed to keep him from running off into danger. He has always been protective of Harry and perhaps in this fanfiction's unique scenario, he believes this extreme measure is necessary to safeguard Harry from some immediate and unseen threat.
It could be that Dumbledore is afraid of what Harry might do if left to his own devices. Harry has a lot of power, and Dumbledore may worry that he could be swayed by the wrong people or ideas. So, in an attempt to keep Harry on the 'right path', he resorts to trying to control him. But this also shows a lack of faith in Harry's character and judgment.
Some writers might do it to create a sense of shock. They want to break the traditional view of the Dumbledore - Harry relationship and get readers' attention. It's a way to be different in the vast sea of fanfiction.
Well, in some fanfictions, Harry might attack Dumbledore if he discovers that Dumbledore has been using him as a pawn in a bigger game all along. This feeling of betrayal could drive him to lash out. Also, if there is a prophecy that he misinterprets to mean Dumbledore is a threat to his future or the future of his friends, that could be a motive.
In a fanfiction, Dumbledore might steal out of desperation. Suppose there was a situation where a curse was about to be unleashed on Hogwarts, and the only way to stop it was by stealing a particular ingredient for an antidote from a place where it was not supposed to be taken. Dumbledore, seeing no other option, decides to steal it. His motive here is to save the school and its students, even if it means breaking the rules.
One possible motive could be his thirst for power. By kidnapping someone with a special ability or knowledge, like a powerful Force - user or a brilliant scientist, he could gain an edge in his quest to rule the galaxy. For example, if he kidnapped a Jedi who knew the location of ancient Sith artifacts, he could use those artifacts to increase his own strength.
In a 'Harry framed Dumbledore' fanfic, a motive could be revenge. Suppose Dumbledore made a decision that Harry thought was wrong and cost someone Harry loved dearly. Harry could then be driven by revenge to frame him. Also, it could be a case of Harry being under the influence of a dark force. The dark force might convince Harry that Dumbledore is a threat and needs to be removed from power, leading Harry to frame him.