How to archive multiple web mail accounts


New Email
Hello. I have multiple email accounts on different web mail services and I view this mail on multiple computers. I would like to archive the older folders from webmail onto a local machine.
a) Is there a way to download the older folders from these web mail services?
b) What file format will they be in? Is there a common format for this, used by all the common webmail providers?
c) If I want to open these archive files in the future, can I just import them into Thunderbird or equivalent? I don't need to actually use Thunderbird to send and receive emails, just to view the folders.
Please note: I understand that IMAP allows a person to read email on multiple computers, but I'm specifically asking if I can keep my webmail and just store certain folders in local files somehow. Thank you.