Step 1: Backing up email directories:
1) Login to cPanel interface for the account in your old server.
2) Open ‘File Manager’ under the section ‘Files’.
3.) Goto /home/user/ (replace user with your cpanel user name)
4.) You can see the mail directory there. Enter into the directory.
5.) Right click on the mail directory and click on the option available to compress the directory (if you wish to backup only emails from specific email or domain, browse to that domain or folder. If the domain is yourdomain.com, look for yourdomain.com and if the email is firstname.lastname@example.org, look for cs directory under yourdomain.com)
6.) Now, download the compressed file to your computer.
Step 2: Restore email backups
1) Login to the cPanel interface of the new server.
2) Create the email accounts just as you did in the old server. You can create email accounts from the ‘Email Accounts’ section under the ‘Mail’ category.
3) Open ‘File Manager’ under the section ‘Files’.
4) Goto /home/user/mail/ (replace user with your cpanel user name we had provided you)
5) Upload the email backup file to the directory.
6) Unzip the backup file and the delete the zip file.
7) Open Webmail and configure it to display the new email folders.