It could be that some of the village elders or leaders send someone to go after her. They might see her running away as a threat to the village's stability or as a potential loss of a valuable asset. Maybe they send a team of ANBU to track her down and bring her back, whether by force or persuasion.
Shikamaru might be sent to find Naruto. He's known for his intelligence and strategic thinking. The village elders might trust him to be able to track Naruto down and either convince him to come back or at least find out his intentions. Shikamaru also has a certain respect for Naruto and would understand his motives to some extent.
Sakura might try to stop Naruto. She is one of his closest friends. She cares about him deeply and would be worried about his safety. Also, she would understand that running away is not the best solution to whatever problems he is facing.
Kakashi could be another one. As Naruto's former sensei, he knows Naruto's capabilities and potential. He would try to stop Naruto from running away because he believes that Naruto has a future in Konoha and running away isn't the solution to whatever problems Naruto is facing.
Perhaps Goku. Goku has a big heart and he's always ready to help those in trouble. He could sense Bulma's distress and come to her aid.
It could be her best friend. In many stories, best friends have a special bond and can sense when something is wrong. So, her best friend might start looking for her and be the first to find her.
He might meet American witches and wizards. Maybe a young and adventurous wizard who shows him around the American magical world.
Well, perhaps the fanfic could be about Sakura being in an unhappy marriage. She might have discovered some dark secrets about her spouse, like they were involved in illegal activities. So, in order to protect herself or her loved ones, she decides to run away. Maybe she has a secret plan to expose her spouse's deeds from afar.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.
Perhaps she is being chased by some unknown enemies in Konoha who are after her for some reason. She might think that running away from Konoha is the only way to protect herself and those she cares about. It could also be that she has a new mission or purpose that she believes can only be achieved outside of Konoha, and she is willing to sacrifice her life in the village for it.
He could go to Hogwarts early. Since he has some connection to the school already through his parents, he might think that the school could be a safe haven. There are also his friends' families there, like the Weasleys, who might take him in.