In many fanfictions where Gil gets kidnapped, it could be for various reasons. Maybe it's by a rival group who wants to use him as a bargaining chip against his friends. For example, in some stories, Uma's crew might be targeted because they are seen as a threat in the Isle of the Lost. Gil, being a part of that crew, gets kidnapped. Writers often use this plot device to create drama and show the loyalty and courage of the other characters as they try to rescue him.
It could be a new character created specifically for the fanfiction. Maybe a young scientist who is working on a project related to the super - soldier serum. Since Bucky has a history with the serum, this would create an interesting dynamic.
It could be a rival gang. They might see Ponyboy as a threat or a way to get back at his own gang. Another possibility is a deranged fan of his, someone who has an unhealthy obsession and wants to keep him all to themselves. Or it could be a group of criminals who mistake Ponyboy for someone else who has something they want.
In the 'darkrai gets kidnapped fanfiction', perhaps a group of trainers who want to control the power of the Legendary Pokémon Darkrai kidnapped it. They might be planning to force Darkrai to battle other Legendaries. This would create a lot of chaos in the Pokémon world as Darkrai is a very powerful and mysterious Pokémon.
Maybe a human who has discovered the existence of vampires and wants to use Renesmee to prove it to the world. They might think that by kidnapping her, they can expose the vampire world and gain fame or fortune. Another option is a rogue werewolf who doesn't like the alliance between the vampires and werewolves and decides to take Renesmee as a way to break that bond.
It could be a new villain who has heard about Tails' mechanical skills. This new baddie might think that by kidnapping Tails, they can force him to build powerful weapons for them.
In such a fanfiction, Tails might be kidnapped by an old enemy like Dr. Eggman. His friends, like Sonic, would then start a rescue mission. They would search all over the place, from the Green Hill Zone to Eggman's secret bases.
In a 'damian gets kidnapped fanfiction', perhaps Damian is kidnapped because he has some special knowledge or power. The kidnappers could be a secret organization. His friends might start to search for him, and along the way, they discover secrets about Damian they never knew before. They have to overcome various obstacles to find him, like decoding mysterious messages left by the kidnappers.
In some 'damian gets kidnapped fanfiction', the kidnapper could be an otherworldly being. Let's say an alien or a magical creature that has an interest in Damian for some strange reason. It could be that Damian has a unique energy or power that the otherworldly being wants to study or harness. Or perhaps it's a human who has been corrupted by some dark magic or technology and is acting on behalf of a more powerful, hidden force that wants Damian for its own nefarious purposes.