an easier method would be to download a full backup from the old server and restore it on the new server.This will automatically create all your email accounts on the new server with same passwords and copy all the old email files. This article becomes relevant when you do not have such a facility or for some reason you do not want to do a full backup-restore.
This is actually quite a simple task if your source and destination accounts are both on cPanel.
Step 1. Create Email accounts on the new server
As a first step, create all your email accounts on the new server, as they exist on the old server. This can be done from your cPanel control panel on the new server. To do this, the iconized link you need to click in your new cpanel is – E-mail accounts
Step 2.Download Emails from the old server.
After establishing FTP connection (using any standard ftp client), navigate to the root directory of your hosting account.
All your emails are under the mail directory. When you drill down to the mail directory,
You need to download everything under the mail directory.
Step 3. Move Emails to the new server
Now, disconnect from the old server and connect to the new server via FTP. Navigate to the mail directory on the new server. Upload everything that you downloaded from the old mail directory on to the new mail directory. This will copy everything on the new server. If you already had some emails on the new server (in case you decided to move emails from old server after having operated the new one for some time), you do not have to worry. They will remain intact and your old mails will simply get added on to the corresponding directories on the new server.
Once all files are uploaded, you are DONE. You can verify that old mails have indeed moved into your newly created email accounts, by logging into your new email accounts using a webmail interface.