I don't think there is a villager cycle you have to go through in City Folk to get a villager back.
However, since Nintendo Wi-fi is no longer in use, you'll just have to wait for that villager to randomly move back in on his own. I don't know if there was a villager adoption thing in ACCF since I never played online on that game, but if there was you won't be able to do it anymore.
If a villager is in boxes on City Folk can't you talk them out of moving away? I did that with Octavian in my CF town when he was in boxes.
I think you're mistaking CF for WW in that case, where you COULD talk villagers out of moving at the last minute when they are in boxes, since in WW they don't tell you about moving beforehand until they already are in boxes.
JVGSjeff has played ACCF for a long time, so I think he knows what he's talking about.