Hello Thomas,
sorry for my late answer.
As Alan was saying, the web server intercepts a request before it serves a page. If it finds a rule that redirects that request, there is no need for the files to exist. So you would manage the rules for the request that you want to redirect on the same server where the old domain is located. If you are redirecting to the same domain, then yes, this means the rule will be on the same server that manages that domain. When somebody looks for your page, then the DNS would point to your server's IP. After that, the request is sent to that IP and the web server will try to serve whatever is needed for that request. But if you successfully add a rule for that specific request (let's say "www.mydomain.com/page1") to be sent to another URL (let's say "www.mydomain.com/newpage"), then the server will redirect that request.