I'm going to be migrating a web server. Right now DNS is handled by the registrar. All I will be doing is changing the A record at the registrar.

DNS should change pretty quick, however I'm concerned about people who have the old DNS cached.

Is there a way on the old server to forward the request to the new IP, but with the same domain?

I'm guessing the answer is no unless I was running the DNS service on the old server.