Perhaps they leave because they discover a new threat that the ninja villages are not taking seriously. Naruto and Sasuke decide to take matters into their own hands. They travel to different places, gathering information and allies. Their relationship also evolves during this time, from just comrades to true brothers who understand each other completely.
In the fanfic, they might leave to train in a secluded place. Away from the distractions of the villages, they can focus on improving their skills. They also train together, combining their techniques and creating new ones. This time away also helps them heal from the traumas they've faced in the past.
One possible reason could be a difference in their goals. Sasuke has always been on a journey of revenge and self - discovery. If in the fanfic, Naruto's path starts to conflict with Sasuke's in a way that Sasuke can't tolerate, he might leave. For example, if Naruto decides to fully side with the village elders whom Sasuke distrusts, Sasuke could see this as a betrayal of their friendship and leave. Another reason could be an external threat that forces Sasuke to leave to protect Naruto. Maybe there's a powerful enemy that specifically targets Sasuke and he doesn't want Naruto to get caught in the crossfire, so he leaves without explanation to keep Naruto safe.
One possible plot could be that Sasuke leaves Sakura to pursue a solo mission of redemption. He feels that his past actions have left a stain on his honor and he must atone for them alone, believing that being with Sakura would only hold him back from his self - imposed penance.
In Shikanaru's fanfic, Naruto might leave Konoha because he feels restricted by the village's rules and traditions. He has always been a bit of an outcast, and leaving could be his way of finding true freedom. The story could follow his adventures in different lands, his attempts to build a new life, and his constant internal struggle between missing his old friends in Konoha and his newfound independence.
One possible plot could be that Dark Naruto leaves Jiraiya because he feels that Jiraiya's teachings are not in line with his new dark path. He might have been influenced by some dark force or his own inner demons. He then goes off to explore his own power, perhaps seeking out other dark entities to team up with, leaving Jiraiya feeling like he has failed as a mentor.
There are fanfics like that. Sasuke might leave Sakura for Karin because of Karin's loyalty. Karin has shown unwavering loyalty to Sasuke throughout the series. She was willing to risk her life for him on multiple occasions. In contrast, Sasuke might think that Sakura's love is more of a childhood crush that she hasn't fully grown out of, and he may be drawn to Karin's more mature and selfless loyalty.
Well, Naruto could leave the village in a fanfic due to a prophecy. He might believe that leaving is the only way to fulfill a greater destiny that the village can't support. As he leaves, he starts to explore different cultures of the ninja world. He learns about new forms of jutsu and how other villages operate. He could also form a team with outcasts from different villages, and together they strive to change the way the ninja world views things.
In this kind of fanfiction, one possibility is that they go on a journey to discover new jutsu together. They might leave the Hidden Leaf Village to find a place where they can train without the constraints of the village rules. Maybe they encounter new and powerful enemies that force them to rely on each other more than ever, strengthening their bond.
One possible plot development could be Naruto leaving to find a way to prevent Sasuke from ever wanting to leave in the first place. He might embark on a journey to discover the root cause of Sasuke's hatred and try to resolve it from afar. This could lead to Naruto meeting new characters who have insights into the Uchiha clan's past and Sasuke's state of mind.
In a 'Naruto Namikaze leaves Konoha' fanfic, it could be that Naruto is framed for a crime he didn't commit. The villagers turn against him, and even his friends start to doubt him. Feeling betrayed and alone, he decides to leave Konoha. As he travels, he meets other outcasts who have also been wrongly accused, and together they form a group to clear their names and seek justice.
Perhaps Naruto in the Anbu has a personal mission that goes against the village's orders. For example, he might befriend someone from an enemy village who is actually innocent but is being targeted by Konoha. When the village refuses to listen to his pleas, he decides to leave. He'll use his Anbu abilities to protect his new friend and find a new place where they can both be safe, away from the political drama of the ninja villages.