1. Create and configure a new email account on your new server with the same email address.
2. Download email files from your old server with FTP. To do this, connect with FTP to your old server, navigate to directory mail/yourdomain.com/emailaccount and download the new and cur directories.
3. Upload files with FTP on your new server (mail/yourdomain.com/emailaccount).
Your new email account will contain all of your emails from the old account, from the old server.