They can only move out when you have 10 or 9 villagers. So, if your 10th moved out, then another will only move in by Streetpass, Campsite, or online interations. If your 9th Villager moved out, then another will only move in by the above methoods, or 4-6 in-game days.
Not that long ago Freya asked if she could move away while Marshal was still unpacking (he had boxes and had not yet left his house) but I don't think this happens to often. So I think you should just wait until the day after they unpack (when they start walking around your town) then check to see who is moving and then if you want them to move and then TT to their boxes date. You could TT like a week forward and I think someone random will be moving out, but I would play it safe especially if you like some of your villagers.
Sometimes I get a villager to ping to move out the day the new is unpacking. If no one pings to move out that day, I TT 3 days and look for a ping. If there is still no ping, I TT one day at a time after that looking for pings. I think this method is actually a little on the conservative side to make sure no wanted villagers move out. I've heard you can actually TT 4-5 days at a time safely, but you should verify this with someone else because I don't have much experience TTing more than 3 days at a time.