I've had AT&T as my an email provider for over ten yrs or so but now I'm thinking of changing services and perhaps going with T-Mobile.

However, as to be expected, I currently have thousands of emails on AT&T I don't want to lose if and when I switch providers. Can anyone recommend a program (preferably free) that would automatically DL and save these emails to my HD so I can archive them for the future?

Most E-mail sites you can login thru Outlook and other PC E-mail clients...
Follow their directions to connect to it. Is important to follow the their direction because many now force PC clients to use encryption via different port(s).

Summery version watch:
POP3 vs IMAP - What's the difference? https://www.youtube.com/watch?v=SBaARws0hy4

If you use POP3... In outlook maybe others... set don't delete (save copy) for X days or never.(forever.) Do This Before connected or will empty the web inbox.
IMAP won't save local copies by default and I forget how to save local copies for e-mail and contacts.

Many e-mail web sites have ways to export contacts and email. search the site.
That may help even if can copy them using local client.

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Dr. Ian Malcolm: Yeah, but your scientists were so preoccupied with whether or not they could, they didn't stop to think if they should.
(Jurassic Park)

The Ogre's Fiero Cave