I use getmail (http://pyropus.ca/software/getmail/) which isn't bound to a specific provider. Currently I backup my gmail and fastmail accounts using this. Here is my daily process:

- mount truecrypt container

- run getmail for each account I want to backup (separate .conf files)

- unmount container

- sync offsite to rsync.net

All of this runs on my raspberry pi 2 and it's really easy to setup (it's practically a bash file with 4 commands).

The output is an mbox file which you can import into Thunderbird or any other client that supports them.

I’ve had good luck using https://github.com/jay0lee/got-your-back to backup gmail.

There are tools such as imapsync, mailsync, etc, which are very useful if you're backing up a remote host which runs IMAP.

I've certainly exported/synced from gmail, gmx, and similar commercial providers to local Maildir archives.

I've been using Gmvault http://gmvault.org for years to back up my Gmail, works great!

I use mbsync (aka isync).

Thank you both! I will look into them.