I decided to switch to a different web host. In the process, I needed to move files, databases, and email from the old server to the new server.
The files were easy. I'd back them up as a .tgz, sftp them down from the old server, and then upload them to the new server. I would've liked to have sftp'd the files directly between servers but the command failed on both.
The databases were easy to move, too. I'd backup the files using either the control panel or the mysql command, download them, and then import them into the new databases I had created on the new server. For a really big file, I'd upload the .sql file to the new server and then use the mysql command line import them into the new database.
The emails were not easy. I discovered that although the retrieval of emails is standard across different email clients, the storage of them on servers is not. The old server used the .mbox format while the new server used the .eml format. Technically, I was supposed to be able to import .mbox files using the Horde interface. When I did this, however, it worked for some folders but not for others.
Here's where Thunderbird comes to the rescue. Connecting via ,imap, I could drop and drag entire directories or wholesale copy them from the old email account to the new one I had set up on the new server in Thunderbird using the ImportExportTools extension.
I tried using Apple's Mail but I couldn't tell when the copying was finished and some of the directories contained thousands of emails. Thunderbird displays a progress bar which helped tremendously in managing expectations about time. Knowing the copy was still in progress also prevented me from tying up threads on the mail server had I started a new copy of emails.