To transfer emails between domains do the following:-

We will call the domain mydomain.com for an example.

1. Make sure you have registered the domain name and it has been added to a cpanel.
2. Using an FTP client, connect to the account where the email accounts are currently and navigate to the etc/ sub folder. There you will see a folder named mydomain.com or the name of the domain you want to transfer the messages for.
3. Download this folder anywhere to your PC. This folder contains the email accounts.
4. Now connect the FTP client to the server where you want to transfer the emails to.
5. Navigate to the etc/ sub folder and upload the downloaded files to the server.

The next steps are to transfer the email messages.

1. Delete the previously downloaded folder from your PC.
2. Simply repeat the process from steps 2-5 above but this time navigate to the mail/ sub folder of the account where the messages are stored.
3. You should be able to open cPanel > Email accounts of the destination account and see the email accounts for mydomain.com listed.