To expand a little on what @simsplayer818 said, you could go into Manage Households and adjust specific settings for neighborhood stories. "Move In" and "Move Out" options are available there, and you could enable or disable them as needed. If you play on a computer version of the game, the mod MCCC also has various options for sims moving in and out that might be worth looking into. Personally I feel that MCCC works much better than any of the vanilla options in the game, though to be fair since my PC version is several years old (and very outdated at this point) I can't really comment on how MCCC would work with the neighborhood stories.
