How do I get my loved one transferred to a different nursing home?

If you are her his or her POA you simply would assure admission to another facility and then give your notice at present facility and do the move.

I talked to the place I wanted MIL moved and they handled notifying her previous place etc

You just do it, but you'd have to have the legal decision-making powers to do that. If Medicaid is involved, I'm sure others will be along to explain that side of things.

I moved my mom from a skilled nursing facility to a memory care closer to where I live. One thing to keep in mind is transferring any supplemental insurance she might have. My mom moved from one county to another and I notified the insurance after the 20th of the month, so we had to pay out of pocket for any doctor visits the first month she was there because her insurance didn't switch over until the first of the month after she moved. It was no big deal, but had I known any changes to the insurance have to be made before the 20th, I probably would have waited another month to move her. We were lucky not to have any real expenses during those four weeks other than the standard intake visit by the facility's doctor.