One common theme is self - discovery. When Harry is exiled, he often has to figure out who he is without the support of his friends and the wizarding world he knew. Another theme is survival. He has to find ways to survive in new and often hostile environments.
One possible scenario could be that Harry is exiled due to false accusations of dark magic. Maybe the Ministry of Magic, fearing his power after the defeat of Voldemort, spreads lies about him being a threat. Another could be that he chooses exile to protect his friends from a new, unknown danger that is targeting him specifically.
It could be a new dark wizard who has risen to power. This wizard might see Harry as a threat to his plans, so he manipulates the situation to get Harry exiled.
One popular 'Harry exiled' fanfiction is 'Harry's Exile Adventure'. It shows Harry being exiled from the wizarding world and his journey to discover new magic in far - off lands. It has great character development and an interesting plot.
One possible reason could be a false accusation of using dark magic. In some fanfictions, the wizarding world can be very paranoid about dark magic, and if Harry was wrongly suspected of delving into it, Gryffindor might exile him to protect their reputation.
One possible plot could be that Harry is framed for something he didn't do. Maybe a powerful dark object is found in his dorm, and the Gryffindor house, fearing for their reputation and safety, decides to exile him. Harry then has to prove his innocence while also dealing with the isolation and new challenges outside of his familiar house.
Obviously, Harry Potter himself is the central character. He is the one experiencing the exile from Britain. Hermione and Ron might also be important. They could either be trying to help Harry from afar or they themselves could be exiled with Harry, depending on the plot of the fanfiction.
One interesting plot could be Harry's struggle to adapt in a new, foreign magical community outside Britain. He might have to learn new magical traditions and make new friends while dealing with the pain of being exiled from his homeland.
He would probably be extremely hurt. Gryffindor was like his family at Hogwarts, so being exiled would be a huge blow. He might initially feel angry and betrayed.
A dark force that has the ability to influence the minds of Gryffindor members. This could be a powerful and ancient magic or a new form of dark magic. It turns the house against Harry without them fully realizing it, making the collective Gryffindor house, under this influence, seem like the antagonist.