It could be that they are involved in some kind of dangerous secret. To protect themselves and the secret, they run away. For example, they might have discovered something about a powerful organization in their world that is not supposed to be known, and running away is their only option to stay alive.
Maybe they are in love and their families oppose their relationship. So, they run away to be together.
Well, they might run away to a secluded forest. There, they could build a small cabin and start a new life away from the chaos of their normal world. Maybe they discover some magical creatures in the forest that help them survive.
One reason could be to escape the constant media attention and pressure that comes with being in the wizarding world's spotlight. Harry especially has always had the eyes of the public on him, and running away might be a way to get some peace.
Maybe they run away because they are tired of the constant danger at Hogwarts. They've faced so many threats, and they just want a break. So they run to a peaceful place where no one knows them.
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.
One possible plot could be that they run away to escape the chaos of the wizarding world. Maybe they are tired of the constant danger and politics. They find a secluded place, perhaps a hidden cottage in the Forbidden Forest. There, they start to build a new life together, learning magic in their own way, away from the prying eyes of Hogwarts and the Ministry. They might also discover new magical creatures and spells in their self - imposed exile.
One reason could be to escape the public spotlight. After the war, Harry is a hero and Ginny is his girlfriend. They might feel overwhelmed by all the attention and want a quiet life, so they run away.
Perhaps in the fanfiction, Harry Potter runs away from Britain because he is disillusioned with the way the Ministry of Magic operates. There could have been a major event where the Ministry fails him or betrays his trust. Also, he may be seeking a place where he can explore his magic without the constraints and expectations that come with being 'The Boy Who Lived' in Britain.
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.