Perhaps Naruto runs away to Suna because he has a secret mission related to Suna that he doesn't want the people in Konoha to know about. He might be trying to uncover a conspiracy that involves both villages and believes that by being in Suna, he can get closer to the truth. So, he leaves Konoha without telling anyone.
One possible plot could be that Naruto runs away to Suna to escape some sort of danger in Konoha. Maybe there's a powerful enemy after him and he thinks Suna can offer him protection. He could also be running away from some internal conflict within Konoha, like issues with the villagers not fully accepting him still.
In this fanfiction, Naruto could run away to Suna because he's seeking a new start. He's always felt a bit of an outcast in Konoha despite his friends. Suna might seem like a place where he can be more anonymous and figure out who he really is without the constant reminders of his past in Konoha. He could meet new friends and allies in Suna and have all sorts of adventures there.
He would surely meet Gaara. Gaara has had his own struggles and he might be able to understand Naruto on a deeper level. They could form an even stronger bond than before, with Gaara helping Naruto hide from those who might be looking for him in Suna.
Suna's reaction would depend on many factors. If Naruto arrives during a time of relative peace between the two villages, they may be more lenient. But if there are existing tensions, it could be a different story. Some Suna ninja might recognize Naruto's potential and try to recruit him for Suna, while others would be worried about the political implications of sheltering a Konoha ninja.
He might face many challenges on his way. In the desert, he could struggle with the harsh environment, like finding water and shelter. Also, he may encounter some Sand Ninja who might be suspicious of him. Maybe he'll meet Gaara and they'll form an unexpected bond, as they both have had difficult childhoods.
One possible plot could be that Naruto and Sasuke run away to find a place where they can train without the interference of the Hidden Villages. They might be on a quest to become stronger together, away from the prying eyes of their respective villages which have different expectations of them.
In this fanfiction, Naruto might run away to Iwa because he feels misunderstood in Konoha. Maybe he discovers some secrets related to his family that make him think Iwa has the answers. He could face many challenges on his way there, like avoiding capture by Konoha ninjas who think he's a traitor.
It could be that a prophecy or a false accusation is made against Naruto, saying he will bring destruction to the village. Hinata doesn't believe it for a second. Instead of waiting for the village to turn on Naruto, she convinces him to run away with her. They then travel to other villages, learning different jutsu and cultures, all while trying to clear Naruto's name.
Maybe he's running away from the pressure in the village. There are always high expectations on him, and he might feel suffocated and need some time alone to figure things out.
One possible plot could be that Naruto feels constantly overshadowed by his family's expectations. Maybe his family is full of powerful shinobi and they expect him to be just as great immediately. He runs away to find his own path and prove himself on his own terms. Along the way, he meets new friends who are also outcasts and together they form a sort of 'family' that accepts him for who he is.
In a 'naruto runs away fanfic', one possible plot could be that Naruto runs away after feeling misunderstood by the village. He might go on a journey to find a place where he can truly belong. Along the way, he could meet new allies or enemies. For example, he might encounter a group of outcasts who are also on the run from different villages. They could team up and form a sort of 'runaway family' where Naruto learns new skills and perspectives outside of the Hidden Leaf's teachings.
One consequence could be that his friends worry about him a great deal. They might go on a search for him, which could put them in danger as well. After all, they care about Naruto deeply and won't just let him disappear without trying to find him.