I've been using a method I saw someone share on google docs, basically the first day, you talk to any amount of villagers but i'm pretty sure you have to talk to the villager you also want to move out. From there the guide said go 15 days in the future but that was just triggering the two week notice that they haven't seen you and I didnt have much success with that.
So i went on to instead reduce it down to 7 days, reset the game and enter it again, talk to everyone and then went on to save and end the game and progress 2-3 days later, went back in to see the thought bubble would pop up with them wanting to move. I had basically every one of my villagers ask me to move out this way but the annoying part is that its random so you just have to hope its one of the villagers that you want to leave lol. If it is a villager you want to stay, i think you just tell him/her to stay and then go in the future a few more days and someone else will ping.
Edit: I only need to kick out Robin and give my friend Audie and then I'm set, from using this method