Another possible reason is for the safety of his loved ones. He may fear that his enemies will target those close to him. By moving away, he hopes to draw the danger away from his friends and family. He might choose a secluded place, like a small cottage in a remote forest, where it's difficult for his foes to find him.
In some fanfictions, Harry Potter might move away because he's disillusioned with the wizarding community. There could have been political unrest or corruption that he doesn't want to be a part of anymore. So he decides to start anew in a different location, perhaps a muggle - dominated area where he can live a simpler life.
One reason could be that he wants to escape the constant attention and pressure. After all he's been through, he might just want a normal life away from the wizarding world's spotlight. Maybe he moves to a far - off place where no one knows who he is.
One possible reason could be that Hermione wants to pursue her own independent magical research. She is highly intelligent and may feel the need to focus on her studies without the distractions that come with being close to Harry. Another reason might be a disagreement over some major decision in the wizarding world. For example, if Harry has a different stance on how to deal with the remaining Death Eaters or how to reform Hogwarts' curriculum, Hermione, with her strong opinions, may choose to distance herself to avoid constant conflict.
Maybe he wants to escape the constant danger in the UK wizarding world. Voldemort's followers might still be a threat even after his defeat, and France could seem like a safe haven.
One possible reason could be that he feels unsafe at Hogwarts. Maybe there are threats from dark wizards within the school that the teachers are unable to fully protect him from. Sirius, being his godfather, represents a safe haven.
She could be running away due to family pressure. Her family might be forcing her into an arranged marriage or pushing her to join a certain dark - leaning group within the Slytherin community.
Perhaps he runs away because he feels misunderstood. In the fanfiction, the other students or even some of the teachers might wrongly accuse him of something. He could be tired of constantly having to defend himself. Like if he was wrongly blamed for a serious incident at Hogwarts, and no one seemed to believe his innocence, he might just leave in frustration.
In fanfic, one reason could be that he might feel constantly in danger at Hogwarts. With all the dark forces targeting him, he could think running away is the only way to keep his friends safe. For example, if Voldemort's followers keep infiltrating the school, Harry might feel responsible and decide to leave to draw the danger away from others.
One reason could be to start anew. After all the turmoil in the British wizarding world, they might want a fresh start away from the memories. The USA offers a different magical landscape with new opportunities.
One reason could be to escape the constant danger at Hogwarts. Maybe there's a new dark wizard threat that the school can't protect him from. So he runs away to find a safer place to hide and train on his own.
One reason could be that he feels misunderstood by the villagers. Despite all he has done for Konoha, some still hold grudges or are afraid of his power. Another might be that he wants to explore the world on his own terms, away from the responsibilities and restrictions of Konoha. Maybe he also feels a sense of loneliness there, as he has lost many important people and the village constantly reminds him of those losses.
Another reason could be that there are some misunderstandings between them. For example, Hermione might misinterpret Harry's actions or words as being too reckless or irresponsible, and she decides to push him away to avoid being dragged into what she perceives as a bad situation. Also, it could be related to her own insecurities. She might feel that she is holding Harry back in some way or that their friendship is becoming too complicated, so she takes the step of pushing him away to simplify things.