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.
It could be that they find out that there's a traitor among the Order of the Phoenix. Feeling unsafe and not knowing who to trust, they run away to figure things out on their own. They start their own investigation to find out the truth about the traitor and how to protect themselves and their cause.
In some fanfictions, they might run away because a new, extremely powerful dark wizard has emerged. This wizard has some sort of connection to Hogwarts that makes it dangerous for them to stay. So they run away to find a way to defeat this new threat, traveling to different magical lands to gather knowledge and power.
One possible reason could be a conflict of values. Hermione is a strong advocate for justice and equality among wizards and non - wizards alike. Draco, on the other hand, was brought up in a family with pure - blood supremacist views. Hermione might run away from Draco in a fanfiction if she realizes that his views are unchangeable and they are fundamentally opposed to her own. For example, if Draco makes a derogatory comment about Muggle - borns in front of her, it could be the last straw and she decides to run away.
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.
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.
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 possible reason could be a growing difference in their life goals. Hermione is highly ambitious and focused on her studies and making a difference in the wizarding world. If Ron is more laid - back and not as driven in the fanfic, it might lead to Hermione walking away. For example, she might be constantly frustrated with his lack of initiative in certain important matters.
Well, it could be about them running away to avoid some dark magic threat at Hogwarts. Maybe a new dark wizard has emerged and they don't trust the school's ability to protect them at that moment.
In a 'Harry and Hermione run away fanfic', they could be running away from the pressure of their pre - destined roles. Harry is always seen as the 'Chosen One' and Hermione as the brilliant sidekick. Fed up with these expectations, they decide to run away and explore the wizarding world on their own terms. They might travel to different wizarding schools across the globe, learning new spells and making new friends outside of Hogwarts.
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.
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.