One reason could be that Naruto feels constantly ostracized in the village despite his efforts to be accepted. In some fanfictions, he might run away because he discovers a dark secret about the village elders that makes him lose faith in the place he once called home.
Well, it could be that Naruto, being the brother of Lucy, ran away perhaps due to some sort of family conflict. Maybe he didn't agree with the way their family was handling things, like their training methods or the expectations placed on them. Another possibility is that he ran away to protect Lucy from some unseen danger. He might have found out something dangerous was coming their way and thought it was better to lead the threat away from her by running off.
One possible reason could be that he felt misunderstood. In the village, Naruto was often ostracized despite his good intentions. Maybe he got tired of trying to prove himself and just ran away. For example, when the villagers still saw him as the nine - tailed fox rather than Naruto, it might have pushed him to leave.
One possible reason could be that he felt misunderstood. In the Naruto series, Naruto was often an outcast in the village. Maybe in this fanfiction, his family didn't understand his dreams or his need for recognition. So he ran away to find a place where he could be himself.
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Naruto could meet the Raikage in this fanfic. The Raikage, being the leader of Kumo, would surely take notice of Naruto's arrival. They might start off on a tense note, as Naruto is from a different village, but later develop an understanding.
It could be due to a family secret. Suppose his family was involved in something dangerous or unethical that Naruto couldn't bear. For example, if they had ties to an enemy organization that Naruto discovered. He is a character with a strong sense of justice, so running away would be a way to distance himself from their actions and figure out how to deal with the situation on his own.
Lucy would likely be very worried about Naruto. She might feel abandoned at first, especially if she doesn't know the reason behind his running away. She could also be in danger if Naruto was protecting her from something. Without him around, she would have to fend for herself or rely on other friends or family members.
It could be that Naruto has some sort of conflict with the people in Konoha. For example, he might have been wrongly accused of something. So, he runs away to Kumo. In Kumo, he might meet new friends and allies who help him prove his innocence and also improve his ninja skills.
For Sasuke, Naruto's departure could be a wake - up call. He might realize how much Naruto actually meant to him and the village. It could make him re - evaluate his own goals and actions, perhaps leading him to try and find Naruto to bring him back.