In theory you could set up a replicated database server in the new location. However replication over the Internet might come with issues like performance and security. Other than that it is difficult to move such a site with zero downtime.
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You can do one thing. After the site is moved to the new server, add a dns forward in the old server.
You can simply change the A record in the zone file of the old server. So even if for someone the nameservers resolve to the old server, it will get directed to the new one.
make sure to leave one of the nameservers as mentioned above to your old host and the other to your new one, that way if it can't find the account on the old one it'll goto the new on, however you would have some lapse between the database sync.