So, one of my villagers just told me that another villager was planning on moving. She told me to go talk to him. So I did. He just keeps talking about Toy Day, and his outfit. Does this mean I have no say if he moves out, and no idea when he will, or...?
Eventually the character that is planning on moving will ping you and tell you so. At that time, you can ask the character to stay or tell it to go. Sometimes the other villagers get the gossip before you do
That character has to "ding" and come up to you before he reveals he is moving.
Save your game then approach the character moving out. He should "ding" at you and walk up to you. Talk to him then and he'll let you know about moving out.