Tire unmount & remount

I just bought some new (used) wheels. I'd like to have them checked for anything unusual (out of round/bends) and then have the rubber on my current wheels unmounted and mounted onto the new wheels and get a good balance. Will many companies will work with used rubber? My tires have a reasonable amount of tread left and I'd like to reuse them. Recommendations?

Regards.

Don't tell them they are used....say they were shaved. The little hole in the wall places (that typically sell used tires) won't care regardless.