I think the easiest method (assuming you don't have amiibo) is to be as friendly as possible to a villager you want to move. It'll make it more likely for them to have the thought bubble of wanting to move.
Alternatively, if you got a cool campsite villager, you can invite them to your island but will randomly choose one of your villagers to replace. This one can be a bit of a pain, especially since there's no guarantee it'll land on the villager you want to kick out.
Then if you have an Animal Crossing amiibo, scan that villager in to camp at your site, fulfill their request three times, and then you can kick the desired villager out. Let me know if you need any other information!