How to quickly transfer a site with corporate mail?

Question

Email

In general, this is the situation. The customer offers a permanent job to support the site on WordPress, but gives a test task to transfer the site from hosting to hosting.
His domain is parked on godaddy. He already gave me hosting accounts. Everything seems to be simple, but ...
He says that people constantly send emails from the site, so any downtime in the process of sending emails is undesirable. And his mail is corporate, that is, if we say his domain is customer.com, then his mail is [email protected]
As far as I understand, he has a mail server somewhere or maybe he has Google corporate mail? I'm in the question of mail and MX records a complete zero.
What should I do so that his emails do not disappear during the transfer and continue to be sent after the transfer? As I understand it, after transferring the site files and database to another hosting and changing the name servers with the registrar, the site should work on the new hosting. And what will happen to the mail, will it work?
Please help clarify the situation.

Answer 1

It also seems to me that the “test task” is the desire to get the result of complex work for free. I came across this in Moscow, and these customers with their “test tasks” hang on avito for months, regularly updating the ad.
I apologize for the offtopic.