There is no direct easy way to do this. Here are the steps that I'd follow
1. Add all clients to new server.
2. Create websites and databases
3. Create Email domains and Email addresses.
4. From old server to new server rsync the /var/www
5. chown -R all the directories to the correct owner
6. Use Imapsync to sysnc the mail accounts one by one http://linux.die.net/man/1/imapsync
If you are using the same mail service (courier, dovecot) Then you could just rsync the mail directory.
Thats the short version.
A very cool option would be to add this as third server to your group and convince till that we need to be able to migrate clients and setting from one server to another. We need this feature no anyways it just doesn't exist.