Why the order of operations is the whole job
The technical steps are not exotic. The sequence is what separates a boring migration from a bad one.
Copy first, while the old system still works
Mail gets copied to the new provider before anything about delivery changes. The old mailboxes carry on receiving throughout. Nothing is at stake yet, because if the copy stalls or the tooling misbehaves, your business has not noticed: mail is still arriving where it always did.
This is the opposite of how it goes wrong. The wrong way is to point the MX records at the new provider first and then start moving data, which means every problem from that moment on is happening to live mail.
Lower the TTL before you need to
Your DNS records have a TTL: how long other servers may cache them before asking again. Leave it high and mail servers all over the world will keep delivering to your old provider well after you switched, because they are working from a remembered answer.
So the TTL comes down ahead of the cutover, giving the old value time to expire everywhere. Then the change propagates quickly instead of dribbling out over days with mail landing in two places. Afterwards it goes back up. It costs nothing and it is skipped constantly.
Switch delivery, then sweep up
The MX records change and new mail starts arriving at the new provider. Then a second copy pass catches everything that landed in the old mailboxes between the first copy and the switch. That delta pass is what makes the difference between "we moved your email" and "we moved most of your email".
Keep the old system until you are sure
The old mailboxes stay, with their data, until the new setup has been running long enough to trust. That is your rollback. A migration without one is a leap of faith with your business correspondence.