Question: how to transfer emails from one hosting account to another?
Answer: Step 1: Create a new email account on the NEW SERVER with the same address and password. You can do this by going into the cPanel on the NEW SERVER and click on Email Accounts.
Step 2: Download the files off of the OLD SERVER. ... 
Step 3: Transfer files to NEW SERVER. ... 
Step 4: Reset Mail Quota. ... 
Step 5: Change DNS. ... 
Step 6: Test.