They should ping you if you're checking in every day and request to move. Also, talk to your other villagers; if someone is planning to move, they'll tell you if you talk to them a lot (several conversations).
they will ping you and then say they want to visit a new place etc... and say they are leaving on a certain date. if you aren't on 'speaking terms' with them (talk to them and you will be!) then a villager you are friends with may gossip to you saying they are moving.