I think it depends on a number of factors.
>Challenger is applying for Gym Membership and has to defeat an existing Gym Leader to take their position
>Gym Leader Retires after a Challenge, and the last known Victor (assuming they take the position takes it)
>The League itself recognizes the new Challenger and probably has no faith in the old Leader
You could go on, I think it really depends. We don't know exactly what happened with Sabrina other than she waltzed in, defeated them, and made her own Gym. Considering she has her own building next door, it stands to reason that the rest of the League recognizes her at the very least, the Fighting Gym probably didn't get many victories or was the first stop for a lot of trainers considering how lackluster it is (but that could be due to poor budget/challengers since Sabrina took over Saffron too). Remember it is Tokyo, realistically Saffron/Celadon would be the first Gyms for many trainers and we do know that teams/levels are set up based on the number of badges in universe. A lot of things could go wrong, we also see Koga advance and be qualified for the Elite Four despite Sabrina and various others being far more qualified, meaning they probably didn't apply, and Koga seems to have some influence thanks to his new status now that Janine is in charge. It's a lot of random shit in reality, since every League functions slightly different in reality to some extent, especially between countries.
