I moved it by moving the database + all files etc., and have done this MANY times (as a web developer I've moved hundreds of sites, changing domain name, and never had a problem!)
However, now that I have moved it, the media library isn't working. It displays errors on upload. I found advice to change to 777 and change back; I used WP advice and changed it to 775 (current).
Now it uploads the images but it doesn't crunch them properly. I can only display full size, it's not generating the other sizes. It errors on every upload (but the files wind up there). Safari isn't showing the files that aren't crunched, but Chrome is. (Weird??)
I know it's a very weird theme appearance, but the media library works properly on the old domain, same version of everything. There is one new post on the new domain which illustrates the problem - the slider takes thumbnail and is displaying a huge picture! Etc.
Why would this have changed with moving domains? It's the same exact server AND FTP user.