
As mentioned in most of the responses above, you need to speak to your villagers on a regular basis. HOWEVER, you also need to give them a chance to ping you. If you speak to them first time you see them on any given day, they will not ping you about moving during the same play session. My personal routine when I play any of my towns, is to just walk around watering flowers, looking for fossils, etc. and make sure any villager who is out and about sees me. I'll go all around the town doing this. If no one pings, then on my second pass around, I'll stop each villager and talk to them. Talk to the villagers who are inside their houses too, as they might tell you someone is thinking of moving.
If someone (inside or out) tells you another is thinking of moving, save/quit and then start again and walk by the potential moving villager to get a ping. They won't ping in the same session when you were told about the move. If you are told about a move during an event day, the potential mover will only ping you the day after the event.
Also, the moving ping isn't necessarily the first ping of the day. Even if some other villager has pinged you for a favour of some kind, a potential mover can still ping you later in the same session.