In case you want to move your site to a new host and wonder how you are going to transfer all user e-mail accounts including folders and messages they have stored in their webmail, here is a quick solution: 

1. Make sure you have updated your DNS records on the old server, lowering the TTL to a few minutes
2. Set up the account (the one from the original server) on your favorite e-mail client, selecting IMAP as the mail protocol
3. Once the remote folders and messages are downloaded on your local machine, copy them to a local data file within the e-mail client
4. Set up the new account on the same e-mail client, again selecting IMAP
5. Copy the folders and messages to the new - empty - account
6. Once you are finished with all accounts, update your DNS settings to point to the new server