One reason could be the loneliness he felt in Konoha. People were often mean to him because he was the Jinchuriki of the Nine - Tails. He might have felt that running away was the only way to escape the constant ostracism and find a place where he could be accepted.
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.
The village would likely be in chaos. Naruto, despite being an outcast in some ways, was still an important part of Konoha. The Hokage and the ninja leaders would probably send out search parties immediately to find him.
Kakashi could be another one. As Naruto's former sensei, he knows Naruto's capabilities and potential. He would try to stop Naruto from running away because he believes that Naruto has a future in Konoha and running away isn't the solution to whatever problems Naruto is facing.
Shikamaru might be sent to find Naruto. He's known for his intelligence and strategic thinking. The village elders might trust him to be able to track Naruto down and either convince him to come back or at least find out his intentions. Shikamaru also has a certain respect for Naruto and would understand his motives to some extent.
One possible reason could be feeling ostracized. In the Naruto world, being different is often not easily accepted. As a female Naruto, she might face even more scrutiny and prejudice. Maybe she's tired of the constant expectations and the way she's treated in Konoha, so she decides to run away to find a place where she can be herself.
Sakura might try to stop Naruto. She is one of his closest friends. She cares about him deeply and would be worried about his safety. Also, she would understand that running away is not the best solution to whatever problems he is facing.
In a Naruto fanfic where Naruto runs away from Kushina, it could be due to various reasons. Maybe he feels misunderstood. For example, Kushina might be too strict with him in the fanfic's storyline, not allowing him to pursue his own ninja way freely, so he decides to run away to find his own path and prove himself.
She could encounter other outcasts. There are likely other ninjas or people in the world who have left their villages for various reasons. By meeting them, she could form a new group, a sort of 'family' of misfits. They could travel together, sharing their skills and stories, and maybe even start a small community away from the traditional ninja villages.
It could be that some of the village elders or leaders send someone to go after her. They might see her running away as a threat to the village's stability or as a potential loss of a valuable asset. Maybe they send a team of ANBU to track her down and bring her back, whether by force or persuasion.