The part people get wrong.
WHAT DOES NOT MOVE
Mailbox contents do not travel with a website backup, unless you take a full cPanel backup from a cPanel source.
WHAT YOU MUST DO
Recreate the accounts here Move the messages Reconfigure devices
MOVING THE MESSAGES
Set up both old and new accounts in a mail client using IMAP.
Drag messages from the old to the new.
Slow for large mailboxes, and reliable.
DO THIS BEFORE THE OLD HOSTING ENDS
Once it is cancelled, the messages are gone.
FOR MANY MAILBOXES
Tools exist to transfer in bulk. Open a ticket and ask.
AFTER MIGRATING
Create every account with the same address Set the same passwords, or tell users the new ones Reconfigure every device
THE DEVICE STEP
Every phone and computer needs the new settings.
Tell users before the change, with instructions.
WHAT TO TEST
Send and receive, both directions, from each account.