Once they are in boxes they won't change their mind. They only way you can get them to change their mind is if you have a relationship with them and they will ask you.
Or send another player in your town to stop them from moving. One of the villagers wouldn't bring up moving the day before so I made a new player. Craig talked to her and got her to stay.
In my guest town I sent a side player that lived close to her house. Stopped the mover and now I wished I let her go.
To change a villagers mind about moving just talk to them you can only do this if a villager ping you and you said you are okay with them moving. I had Nan that wanted to move and I said I'm okay with that then I talked to her and she offer me if she wanted to move again.
♥ Try to let them ping you before they are in boxes,
then you'll be able to change their minds c:
If you heard rumors about a villager you want to move is moving,
don't talk to them even if they ping you,
they might change their minds and not move. ♥ ~♚Vickie