Another possibility is that they want to explore the wider magical world. They've been so focused on the problems in Britain that they realize there's a whole world out there. They leave to have adventures, learn new magic, and meet different magical cultures in other countries.
Perhaps they're disillusioned with the wizarding government in Britain. They may have seen too much corruption or inefficiency during and after the war, and they decide they can't be part of that system anymore. So, they leave Britain to find a more just magical society elsewhere.
One reason could be the media attention. After all they've been through, they might be tired of constantly being in the spotlight. The British wizarding press could be hounding them, so they leave to get some privacy.
One reason could be to escape the constant reminders of his past traumas in Britain. The place is filled with memories of Voldemort and the battles he fought. Maybe in the fanfiction, he just wants a fresh start away from all that chaos and danger.
One reason could be to escape the lingering dark magic and the constant reminders of their past battles. Britain, especially Hogwarts, has seen so much turmoil that they might feel the need for a fresh start. Another reason could be to explore the wider wizarding world. They've been mostly centered around Britain in the original stories, and in fanfiction, they could leave to discover new magical cultures and spells in other parts of the world.
One reason could be to search for a powerful magical artifact outside of Hogwarts that they believe is crucial for defeating a new dark threat. In fanfiction, they might feel that Hogwarts has become too dangerous or restricted in terms of resources for their mission.
One reason could be to find peace. After all the chaos and war in England, they might just want to get away from it all and start fresh somewhere else.
One possible reason could be a misunderstanding. Maybe they misinterpreted something Hermione said or did, and in a fit of anger or confusion, they decided to leave. For example, if Hermione was working on a project alone without consulting them, they might have felt left out and left in a huff.
In fanfiction, Harry could flee Britain due to a sense of disillusionment with the wizarding world there. He has been through so much trauma and has seen the flaws in the system. For example, the way the Ministry handled the threat of Voldemort in the past was often inept. He might also be seeking a place where he can explore magic in a different way, free from the traditional constraints of the British wizarding community. He may want to start anew in a place where no one knows him as 'The Boy Who Lived' and can just be Harry, an ordinary wizard exploring his magical abilities.
In some fanfiction, Harry leaves to seek more powerful magic. He realizes that the magic in Britain is somewhat limited in the fight against Voldemort. So he travels to other places, like ancient magic - rich regions in Asia or Africa, to find a way to defeat Voldemort. He hopes that by leaving, he can gain the upper hand in the battle.
One possible reason could be to seek new opportunities. In fanfiction, the States might be depicted as a place full of exciting prospects in various fields like magic (if it's a Harry Potter - related fanfic) or just in general life. For example, he could be interested in exploring different magical cultures there.
She could be frustrated with the Ministry of Magic in Britain. Maybe they're not taking Voldemort seriously enough or are even hindering her attempts to fight him. So, she decides to go directly to Voldemort to end it all on her own terms.
There's 'Beyond the British Isles' where they leave because they feel ostracized by the wizarding community in Britain despite their heroics. They head to America and get involved in the American wizarding world, facing new challenges and making new friends.