Well, in such a fanfiction, Sakura might be used as a bargaining chip. The Akatsuki could try to exchange her for something valuable, like secret jutsu. Or they might want to extract information from her about Konoha's defenses.
Well, one possible plot could be that the Akatsuki kidnap Hinata to use her Byakugan for their own purposes. Maybe they want to study it to gain an edge in their battles or espionage. They might hold her captive in one of their secret hideouts, which could be an old, abandoned fortress.
The kidnapped character could use their own hidden jutsu skills to break free. For example, if they have a teleportation jutsu they've been hiding, they could use it when Sasuke is distracted. Another way is to appeal to Sasuke's emotions. If they can remind him of his better nature or a past event that made him more empathetic, he might let them go.
They might treat her harshly at first to break her spirit. But then, realizing her value, they could start to offer her deals, like freedom in exchange for her cooperation. However, Sakura is a tough character, so she'd likely resist their attempts.
If Sakura gets kidnapped by the Akatsuki in fanfiction, Sasuke could also get involved. Since he has his own history with the Akatsuki, he might use his knowledge to track them down. Also, Sakura herself could be trying to find a way to escape from within. She's a strong kunoichi, after all. She might be gathering information about the Akatsuki's plans while being held captive.
Kakuzu might also be a main antagonist in this fanfiction. He's always after money and power. Maybe he sees Sakura as a valuable asset, either to sell her back to Konoha for a large sum or to use her skills for his own gain. He's not very empathetic and would likely be ruthless in his treatment of her.
In fanfiction, if Naruto gets kidnapped by Akatsuki, he might be used as a bargaining chip. The Akatsuki could try to exchange him for something valuable like the Nine - Tailed Fox's chakra. Or they might try to extract the Nine - Tails from him forcefully. Naruto would surely put up a fight though, using his Rasengan and other jutsu.
There's also a possibility that Hinata's own clan, the Hyuga, would take matters into their own hands. They have their own unique techniques and a sense of pride. They would not allow the Akatsuki to keep one of their own captive for long and would send their strongest members to retrieve Hinata.
In fanfiction, if Sasuke and Sakura joined Akatsuki, it could lead to a complete shift in the power dynamics. Sasuke already has a complex relationship with the Akatsuki members. With Sakura by his side, she might act as a moral compass for him. Maybe they would try to reform the Akatsuki from within, or they could be forced to carry out some dark missions that would test their relationship.
In a fanfiction, Naruto and Sasuke in Akatsuki might have a story where they are sent on a special mission together by the Akatsuki leaders. Along the way, they face many challenges that test their bond. Sasuke might initially be more in tune with the Akatsuki's ways, but Naruto tries to convince him to find a better path. As they complete the mission, they start to change the Akatsuki's ideology bit by bit.