I will kick the ones that are my least favorite among my other villagers first, but i have to get their framed pictures before doing that. In my case they're Lyman and Erik, but since Erik just recently moved to my island and i haven't got his picture so i have to write him off from the list and replace him with Norma (Although i actually love her)
And about inviting new villagers, i only rely on Campsite and Island Hopping. I only invite those who are personally my favorites in Campsite, and i only do Island Hopping when certain villager move out and i need to find the replacement and I'll take any villager who comes first