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Your best bet is to maintain the same pages in both locations for a few weeks after you have updated DNS.
Assuming you have the option of keeping the site up on your old site, do so until your logs show it's not getting any more traffic. Most people's DNS will have been updated within a day or so of the change taking effect at your registrar ... but don't turn off the old site until the traffic to it is down to a small enough trickle that you don't mind losing those visitors.