When a villager is in boxes, you go to their town and talk to them. They will say something along the lines of, "I'm moving but I don't know where I'm going..." and it gives you the option to invite them into your town as long as you have space for them and they're not in your 16 villager cycle.
Haha sorry about the typo XD I use an iPad to post so I guess it auto corrected.
But wow I didn't know you could do that!! How cool!!!!
Thank you for letting me know this fabulous information!