They may or may not. Personally, I'd wait until after the bloom season or preferably, late summer/early fall.

If you "have to" move it, water it the night before & try to move the rootball without breaking it and include as much of the tiny fibrous roots as you can. They absorb moisture thru those small roots. My azalea Cheers did not bloom as profusely when moved several winters ago but a native azalea did fine when moved before the end of the year. Winters here can be mild so Cheers may have been almost awake when I moved it. I do not think it had leafed out when moved as I would have left it alone in that case. But the native one was dormant and had to be moved due to work being done on the roof. It was a pain to move and required two people.

Is there anything wrong with moving them in early winter/late fall?

I have a couple I would like to move. I can't remember which kind they are, but one of the popular series from a big box store.

"early winter/late fall " sounds like a good choice. You can plant new ones too during that time in mild winter locations. Big boxes typically carry either sasanqua (shorter leaves, most bloom early to late Fall... with exceptions) and japonicas (larger leaves, most bloom late Fall to Spring... with exceptions as usual). Try to move as much of the small/tiny fibrous roots near the top 4" (or so) as you can. Japonica leaves require shade after 11am or so. Sasanqua leaves can usually take full sun but I still plant them in morning only or dappled sun locations.

thanks, luis!

I'm moving them to more shade. They are doing 'ok' where they are now, but I'm sure they'd be happier elsewhere.