Well, Naruto could mimic Kakashi's calm and collected demeanor during fights. Kakashi is known for not getting flustered easily. Naruto might start to hold back his usual exuberance and be more stealthy, like Kakashi sneaking up on enemies. He could also try to use a variety of jutsu in quick succession, similar to how Kakashi combines different techniques smoothly in battle.
Well, Kakashi might introduce Naruto to more strategic ways of fighting. For example, he could teach Naruto how to better analyze his opponents before attacking. Naruto, who was previously more impulsive, may start to hold back and look for the right moment to strike, like Kakashi does with his Sharingan.
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.
Kakashi's training might make Naruto more strategic. For example, instead of just relying on brute force with his Rasengan, he could start using it in combination with feints and misdirections. If Kakashi teaches him how to read an opponent's movements better, Naruto can time his attacks more precisely.
Her fighting style might be more refined. Kakashi is a very technical fighter, so he would likely instill in her a sense of precision. Unlike the male Naruto who often relied on brute force in the beginning, she might start with more calculated moves. For example, she could be better at using kunai and shuriken with deadly accuracy from the start.
Naruto's fighting style could become more versatile. He already has a wide range of skills, and adding the Chidori gives him another option. He can now combine it with his shadow clones. For example, he could send clones to distract the enemy while he sneaks in with the Chidori.
Often, Gai is paired with Kakashi in Hurt Kakashi fanfiction. Gai, with his unwavering belief in friendship and his eternal rivalry with Kakashi, can be a great character to interact with a hurt Kakashi. He might try to pull Kakashi out of his slump through their friendly competition. Also, Rin is sometimes included. Since she has a significant past with Kakashi, her presence in the fanfiction can either be a source of comfort or more pain, depending on how the story is written.
In such fanfiction, the Kakashi OC often has unique physical features. For example, they might have distinct hair color or eye shape that sets them apart.
Kakashi might be initially surprised. But then, he would probably use it as a teaching moment for Naruto. He could say something like 'Naruto, stealing is not the way of a ninja.' And then give him a punishment like extra training.
Often, Naruto is paired with Kakashi in such fanfictions. Their bond from the original series can be explored in new ways in the context of Kakashi's reincarnation. For example, Naruto might be the one to discover Kakashi's reincarnated form and help him adjust.