It could be out of envy. Naruto has always been able to make friends easily and has a large support network. Sasuke, who has had a more solitary and troubled life, might want to strip Naruto of his freedom as a form of revenge for having what he feels he lacks. So, he forces Naruto to be his pet to assert his own dominance over Naruto's more 'fortunate' situation.
One possible motive could be power - related. Sasuke might see Naruto as a symbol of something he wants to control, maybe Naruto's power or his unwavering spirit. By making Naruto his pet, he could feel a sense of superiority.
Maybe it's a result of some dark turn in Sasuke's character development in this fanfiction. He could be influenced by some external force or his own inner demons. He might think that by making Naruto his pet, he can isolate Naruto from others and have Naruto solely focused on him. This could be a twisted way of getting Naruto's attention or a form of self - gratification in terms of having power over someone he has a complex relationship with.
One possible motive could be if Sasuke was on a path of destruction that endangered the entire ninja world. Naruto, as the protector of peace, might see killing Sasuke as the only way to stop him. For instance, if Sasuke was about to unleash a powerful and uncontrollable jutsu that would wipe out many innocent lives.
One possible motive could be if Sasuke goes too far in his pursuit of power and becomes a threat to the entire ninja world that even Naruto can't reason with. Maybe Sasuke starts to kill innocent people without hesitation, and Naruto, as the protector of the village and the one who believes in peace, sees no other option but to stop him by killing. Another motive could be some kind of dark magic or mind control that makes Sasuke an uncontrollable force of destruction, forcing Naruto to take drastic measures.
It could also be a power play. In the Naruto universe, power is a big deal. Some might frame Naruto to get him out of the way in their pursuit of power. Since Sasuke has his own group of fans in the fic, they could be used as tools to achieve this by making it seem like Sasuke is against Naruto, when in fact it's all a setup.
It might be for Naruto's own good. Sasuke may believe that Naruto is on a dangerous path that will lead to his downfall. So he captures him to keep him from making bad decisions, like sacrificing himself for the village or something similar. He wants to save Naruto from himself in a sense.
Perhaps Fem Sasuke in this fanfiction has a different view of the world. She might believe that Naruto is a threat to her goals or to the future of the ninja world she envisions. It could also be that she has some deep - seated emotional pain that she blames Naruto for, and that leads her to kill him.
Perhaps they are being manipulated by an outside force. An enemy of Konoha could deceive them into believing that the village is a threat to the greater good. This would lead them to take the drastic step of destruction. For example, if they are shown false evidence of Konoha planning to start a war that would cause mass destruction, they might act to stop it in what they think is the only way - by destroying Konoha.
Another reason could be related to their different ideologies. Naruto is all about friendship and protecting the village at all costs, while Sasuke has his own complex views which might lead him to think that Naruto's way is wrong. This clash of beliefs could escalate to a point where Sasuke believes that slaying Naruto is the only way to impose his own ideology.
One possible motive could be to force Naruto to understand his own power. Sasuke might think that by kidnapping Naruto, he can push Naruto to a situation where he has to tap into his full potential. Maybe Sasuke feels that Naruto has been relying too much on his friends and needs this shock to grow stronger on his own.
One possible motive could be jealousy. Maybe the sister is jealous of Naruto's popularity, his strength, or the attention he gets from others in the village.
One possible motive could be if his brother was somehow a threat to the village. Naruto is very loyal to Konoha, so if his brother was planning to destroy it or harm its people, Naruto might feel compelled to stop him at all costs.