I would first copy the site before transfering the domain. That way you have the time to test your new home and configure all the necessary things. Once you're sure that everything is set up correctly, pull the domain switch, and 48 hours later all your visitors will see the new pages.
Only if the content of your site changes many times a day, this procudure might lead to problems. I just moved my weblog to TCH and did the following:
Bought a hosting plan;
Disable comments, so no changes could be made to my site;
Asked the support staff to transfer my site. Since I already used cPanel on my old site, this only took 10 minutes;
Checked everything;
Enabled comments on my new site;
Changed the domain information.
Sit back and watch the visitors arrive to my new home.