The safest migration process copies the website to the new environment before changing DNS. Test the application using a temporary hostname or local hosts-file mapping so problems can be fixed while the existing site remains live.
What to consider
For dynamic websites, plan a final database or file synchronization close to the cutover. Lower DNS TTL in advance where appropriate and avoid unrelated changes during the migration window.
After DNS changes, verify website functions, forms, email, SSL and scheduled jobs. Keep the old environment available temporarily until traffic and data are confirmed on the new platform.