This has happened a few times for me, but the two most prominent in my mind are Eugene and Mott.
I had Eugene living in my main town, Haunton, and his design is one of my favorites. He moved out unexpectedly, not telling me and apparently not telling any of my other villagers, when I was trying to get Fuchsia to move out and make room for Whitney.
My heart sank when I saw Isabelle tell me that he was moving out, and I was really bummed out for like an hour. I bounced back though and soon after got Fuchsia to finally move out. I didn't end up liking Whitney since her eyes look weird when you talk to her straight on, so it was kind of a waste. It kind of worked out though in the end, because I ended up replacing those two spots with Beardo and Stitches.
Despite getting over it quickly and bouncing back, I still really like him and hope to acquire him again in one of my other towns.
I had Mott living in one of my other towns and was trying to acquire dream villagers. I liked him but liked my dream villagers more, so when he asked to leave, I accepted his request. I came to regret this very quickly though, because I came to really like his design and that his name made me think of apple sauce.

It was too late to stop him from moving out by the time I really wanted to keep him, as I think I was adopting some other villager at the time.
Again, like Eugene, I plan to reacquire him in the future.
I guess you could say that since I want to get them back, that I haven't exactly "moved on", but I'm not distraught or anything not having them for the time being. A couple examples of some villagers I had to get rid of that I liked but that didn't fit my town themes would include Agent S and Tucker. This might also be the fate of Lucha and Gladys.