The village elders could be antagonists. They might see Naruto and Kakashi as threats to their traditional way of ruling the village and use false accusations to banish them.
In this fanfiction, Naruto and Kakashi are banished because of a prophecy that some in the village misinterpret. The prophecy seems to suggest they will bring doom, but in reality, it's about a different kind of change. While banished, they explore different lands, learn new jutsu from other ninja clans, and eventually return to the village stronger and wiser, proving the prophecy was misread.
One possible plot could be that Naruto is wrongly accused of a serious crime in the village and is banished as a result. Feeling betrayed, he decides to confront Kakashi, who he may see as representing the village's unjust decision. The fight could be a way for Naruto to vent his anger and also prove his innocence.
There are several fanfiction websites where you might find such a story. Archive of Our Own (AO3) is a popular one. Just go there and use the search bar with relevant keywords like 'Naruto banished fights Kakashi'. Another option could be FanFiction.net. However, it might take some time to sift through the results to find exactly what you're looking for.
Naruto might also initially feel a sense of abandonment. However, his kind - hearted nature could lead him to try and see the good in the situation. He could start to explore the world outside, learning about different cultures and ninja techniques. He may use this experience to become more empathetic towards others who have also faced hardships, and in the long run, this banishment could be a turning point in his life to become an even greater ninja.
At first, Naruto is probably shocked and confused. He might not understand why the village would take such a harsh step just for hurting Sasuke. He may feel betrayed, especially if he thought he had friends and allies in the village.
Well, there could be. Fanfiction writers love to explore unique and unexpected relationships. Since Naruto and Kakashi are both popular characters in the Naruto universe, some writers might come up with such a plot for the sake of creativity and exploring different storylines. But it's not a part of the official Naruto story.
A possible plot for this type of fanfiction could involve the banished ninja having a secret mission that the village doesn't want to be associated with. So, they banish the ninja to make it seem like they have no part in it. The ninja then has to complete the mission while also dealing with the stigma of being banished.
Well, in a fanfiction like this, it could explore Naruto's sage mode abilities in a new context while he's with Yugito who has also been banished. Maybe they form an alliance to fight against the forces that banished them. They could discover new powers together and change the shinobi world's view of them.