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Yesterday my favourite villager had a thought bubble about moving islands. I told them I didn't want them to and they said they would stay, but am now wondering will this villager eventually move now that they have had the thought? What is the cycle when it comes to villagers moving, and is it just a matter of time kind of thing, or will they now stay? Also, can villagers move without us saying they can/talking to us?
Villagers can only move if you give them permission to move. That particular villager of yours won't move out, unless the next time they ask, you grant them permission. There's really no cycle when it comes to villagers moving. It seems like if you have a strong friendship and have given medicine to them when they were sick in the past, it becomes less likely for them to want to move out. Just an observation of mine from a ridiculous amount of time spent moving out villagers.
It really seems to be random which villager gets the thought bubble to move out.
If you have a higher friendship level with a villager, you're more likely to get thought bubbles followed by a gift versus a villager who just moved in, but if the game decides it's time to ping someone to move out, it really seems to be random which villager it is. That is, if you TT and go to a date in the future and someone asks to move out and you don't save, go back to your original time/date, and go back to that same future date, a different villager will ask to leave.
That is to say, don't take it personally if one of your favorites asks to leave. But as has already been mentioned, they can't leave without your permission.
As to predict when a villager will ask to move out, it seems to be around every 14 days, give or take, with a level of randomness involved.