What needs to move?
Usually the files, databases, application configuration, cron jobs, redirects, SSL certificate and form e-mail settings need to be migrated.
Testing before DNS change
Before changing records, the site should be tested on the new hosting account, with a hosts-file preview or a temporary test address.
After the switch
After DNS change, check the site from several networks, forms, redirects, mixed content, SSL certificate and application error logs.