My interpretation was IMAP (or at least POP configured to leave messages on server) combined with local folders, but good call with download all messages.
Local folders are a work of Stan, but if you have to have them the mail client should have the decency to use something portable like mbox[1] files. Obviously Microsoft are prime offenders for doing their own thing for no obvious reason.
[1] mbox is literally a load of emails, in plain text as they're sent on the wire, concatenated together. Mail clients (and indeed IMAP servers) usually present them as a folder. It's not particularly fast, but there's no shortage of software to read them, and it's easy enough to write your own if you have to.