in the case of Adobe BC you would give the new site a name like (yoursitename.businesscatalyst.com) and the transfer whatever.com to this new site so;
whatever.com & yoursitename.businesscatalyst.com would both go to the same place... allow a week for google search to catch up
If your client doesn't want to keep the old whatever.com name then things are easy but its not a major stress
Unless things have changed since I used Squarespace, you would not be able to upload a Muse site. Squarespace is a self-contained system, similar to Adobe Portfolio.
You will need to use BC or a third party and, as USS Norway says, transfer the domain name. You'll need to find out where the domain is registered and change the DNS settings to point to the new host. Most providers have comprehensive documentation on how to do this.
Thanks David. Your advice confirms what I was figuring out from all the documentation (and lack there of) on how to upload Muse sight. And I will know for next time.
Do you know if WIX is self contained as well? I work with a lot of small businesses that launch on Square and Wix before they give up and come to me.
Yes, Wix is also self-contained.