Help Me!!! Confused about cycling

Alright. I need to know if one can use amiibo villagers to cycle out 16 villagers. Here's why. One of my dreamies, whose card I unfortunately do not possess, is being held in my other town. I understand that 16 villagers must cycle through before I can move them back in. Here's my dilemma. I do, in fact, own about 20 amiibo cards. I'm wondering if I can move in 16 of them to cycle with. Will they show up in main street? I'm worried they won't because if you can move them in whenever without cycling, then maybe they simply aren't included in the cycling and they won't count towards my 16 villagers no matter how many I move in and out. Could somebody PLEASE confirm something so I don't waste time cycling amiibos??? Much appreciated in advance.

I've heard this works as long as the 16th villager that moves out is a natural move out, so you can invite them from another village

Thank you! I'll keep cycling the amiibos then. I just really want Fauna back.