I followed this from a guide by pwnapples on youtube and it works. You talk to the villager once and then ignore them the entire time they are living in the town. If they try to run up to talk to you ignore them or if you accidentally talk to them soft reset your game. This method has worked for me. But if you are trying to keep all your other villagers as well be patient.
More than likely you have built a good relationship with the ones you want to keep so they will also want to move out.
Talking to the villager you want to move out once and then ignoring is much faster than befriending them and expecting to move out. Although it is a lot more frustrating ignoring them in case you find a lost item or if a villager wants you to complete a delivery to that villager you want moved out.
Good luck! Because it is a pain in the catoosh.
Also, if you have 10 villagers there is no way to move out the last villager who moved in unless you move out the other 9 villagers. So let's say you had 9 villagers and then you accidentally (or purposely from the campsite) got a 10th villager, there is no way to move that 10th villager out. It is impossible (I have tried), you must move out one of the other 9 villagers to get that one annoying 10th villager to move out.