Perhaps he is on a quest for self - discovery. Jon Snow has always had a complex identity, and in fanfiction, he may feel that he can't fully understand himself while remaining in Winterfell. He runs away to explore the world outside, meet new people, and learn more about his true nature, which he hopes will ultimately make him a better leader or person.
Arya might be inspired to go after him. In fanfiction, she could see his departure as a call to adventure herself. She might feel that there are things out there that she can learn from Jon or things she can do to help him. His leaving could be a catalyst for her own journey outside of Winterfell.
He might first find a place to hide, like an abandoned building. There, he starts to realize how difficult it is to be on his own. He has to find food and water by himself.
Perhaps Naruto feels that Kushina is being overprotective. He wants to prove his independence and go on his own adventures without her constant supervision, so he decides to run away.
He might miss out on some crucial training. Jiraiya has a lot of knowledge and skills to pass on, and by running away, Naruto could fall behind in his development as a ninja.
One possible reason could be that Naruto might be afraid of the intense training Jiraiya has in store for him. Jiraiya is known for his strict and grueling training methods, and Naruto, being young and sometimes impatient, might just want to escape that for a while.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.
One possible plot could be that Naruto feels constantly overshadowed by his family's expectations. Maybe his family is full of powerful shinobi and they expect him to be just as great immediately. He runs away to find his own path and prove himself on his own terms. Along the way, he meets new friends who are also outcasts and together they form a sort of 'family' that accepts him for who he is.
One possible reason could be that he feels misunderstood. In the village, he has always faced discrimination because of the Nine - Tails inside him. People often look at him with fear and suspicion. So, he might run away to find a place where he can be accepted for who he is.
One possible reason could be that Peter might feel like he's a burden to Tony. Maybe he made a big mistake and is afraid of Tony's reaction. For example, if he accidentally endangered the lives of others during a mission while under Tony's supervision.