As far as I know, the license is linked to your domain, regardless of who hosts it. In case you move to a new accommodation, you are not, in effect, doing anything to the real domain, it still works as it was. Moving the content means copying all the files with the correct permissions: compress and unzip it all, and move the database through which means export and import again. The controls should be done with your new username and password of the database, obviously.
When you move domains, or change nameservers, etc., there may be a 24-48 hour period in which your site can be on and offline before it is established.
Moving your site to a new domain name (as I did very recently) will mean downloading your database and then searching and replacing the old domain URL with the new one before importing it, along with any other manual modification within your files, in case you have made any. A great program for this is a free program called TextCrawler.