The best solution
The best solution would be an IMAP based email setup, such as you can get for free with Gmail. All your email is stored "in the cloud", but you can access it with any email client that can handle IMAP. So Outlook, Thunderbird, and probably about two dozen others at least.
Moving a mailbox file around is at least a possibility, but it does get a bit impractical for more than a short term solution, and there could be some kind of unforeseen consequences down the road if you ever upgrade Outlook on one system but not the other, or one of a dozen other possibilities comes to pass.