Here is a tip from the Delicious Brains team for when you are migrating a site to new hosting. They suggest setting up the old site to forward to the new site when you change the DNS, in case there are some DNS servers that are slow to propagate or people have the old site cached. They provide instructions for Nginx and Apache.

DNS Proxy Migrated Sites
Mike Davey @ deliciousbrains.com • 8 months ago