/etc/letsencrypt/live is useless on it’s own as it only contains symlinks. For a successful migration you’ll need to copy the relevant subdirectories of live/ and archive/ along with the relevant .conf file in renewal/
There are several ways to do this, depending on your exact requirements.
The easiest is simply copy the certificate, private key and CA cert over to the new server, then install a new copy of certbot ( or whatever client you use) in generate a new certificate when it’s time for renewal.
If the certificate covers multiple domains, it becomes a little more complex, but please describe if you want it to be a separate cert or whatever, and we can provide more info.
The reason I wanna copy the files for that domain to the new server is because it takes a while for the A records to propagate to the new server and until they do so, letsencrypt can’t generate a new cert on the new server. (Plus, I may have created the older cert less than a week ago.)
Let’s Encrypt doesn’t rely on DNS caches as a client would, so it will be willing to issue to the new web server as soon as the authoritative DNS servers for that name begin giving the new answer. Still, the plan to just copy across certificates and accompanying keys makes sense.