EnterraHost
Moving mail is two separate jobs that people tend to think of as one. Copying what already exists, and switching where new mail is delivered. Doing them in that order is what keeps the move boring.
Copying what you already have
Mail is copied over IMAP, which is the same protocol your current client uses to read it. Calendar and contacts come across too on the tier that carries them, so the move is not mail-only on Pro.
The copying happens while your current mail keeps working. Nothing needs to stop, and you can keep reading and sending from the old provider throughout. That matters if you are moving a business address rather than a personal one.
Switching new mail
New mail follows your MX records, so the switch happens when those change, not when the copying finishes. That is the moment mail starts arriving at the new place, and it is the only part with a hard edge to it.
Because DNS has to propagate, the switch is gradual rather than instant, because a resolver keeps the answer it already holds until its copy expires, so for a period some senders reach the old provider and some reach the new one. That is normal, and it is why the copy should be finished first.
The order that avoids trouble
Create the mailboxes, copy the existing mail, confirm what arrived, and only then change the MX records. Doing the DNS change first loses whatever arrives in between, because the new mailboxes hold nothing yet.
If you are running your own mail server behind this, the MX change is also where keeping filtering while you migrate becomes relevant, since the filter has to be told about the new destination.
What does not come across
Mail that only ever existed on one device and was never on the server cannot be copied by a server-side move. Neither can anything already deleted. If either matters, check before the old account is closed rather than after.
Leave the old provider running for a while after the switch. It costs a little, and it is the only way to see mail that is still arriving there because a sender has not seen the new records yet.