It could also be due to a particularly bad encounter with someone in the village. For instance, if he was bullied severely by some older kids and he felt that there was no one to protect him. He might have thought that running away was his only option to avoid further harm.
Mal might run away in the fanfiction because she's constantly being judged for her past as a villain's daughter. Despite her efforts to fit in Auradon, there are still those who look at her with suspicion. She could also be running away from a future that she doesn't want. For example, if there are plans for her to take on a role she's not ready for or doesn't believe in, she may choose to escape.
One possible reason could be Naruto's internal conflict. In yaoi fanfiction, the relationship is often complex. Naruto might be running away because he's confused about his feelings for Sasuke. He may not be ready to face the intensity of the relationship that the yaoi concept implies.
One reason could be that some in the village feared his potential as the Jinchuriki of the Nine - Tails. They might have thought locking him away would prevent him from causing chaos. Another reason could be related to those who still held grudges against the Fourth Hokage for sealing the Nine - Tails in Naruto. They took it out on Naruto by imprisoning him.
One consequence could be that his friends worry about him a great deal. They might go on a search for him, which could put them in danger as well. After all, they care about Naruto deeply and won't just let him disappear without trying to find him.
One possible reason is she's had an argument with her friends and feels she needs to get away to cool off.
Another possible reason is a sense of not belonging. Maybe she is different from the rest of her family in terms of beliefs or interests, and she runs away to find a place where she can be herself without judgment.
Maybe he's tired of his restricted life. In the story, he might be feeling suffocated by all the rules and expectations placed on him, so he decides to run away to find freedom.
One possible reason could be that he discovers a dark secret about his family. Maybe he finds out that his parents were involved in something he can't accept, like a deal with a dangerous enemy. So, to avoid being associated with it, he runs away.
One possible reason could be that Naruto feels like an outcast. Despite being part of Team 7, he might think that Sasuke and Sakura are closer to each other. Naruto has always been alone before joining the team, and if he feels excluded again, it could lead to him running away. Also, he might be afraid of endangering his teammates. Naruto often attracts trouble because of the Nine - Tails inside him. He may think that by leaving, he can keep his friends safe from the enemies who are after him for the power of the Nine - Tails.
Another reason could be that Naruto is being targeted by a powerful enemy within Konoha. He doesn't want to put his friends and the village at risk, so he runs away to draw the enemy away from Konoha. This would show his selfless side, sacrificing his own safety for the greater good of the village.