How are you accessing them? Look in your account settings in your current email client and the name of the incoming mail server will typically have IMAP in its name if you're using IMAP and POP if you're using POP.
If you're using IMAP, then you don't have to do anything other than to set up your account on your new machine in the email client of your choosing. All of your mail is kept on the IMAP server and it's fetched when you open a message to read it.
If you're using POP, then a bit more detail would be needed about how your account is currently set up under WLM for POP access. Things become more complicated because of how POP works and whether you've changed any defaults with regard to how quickly (or slowly) you're purging messages from the server.
Brian - Windows 10 Pro, 64-Bit, Version 20H2, Build 19042
There are many who labor under the gross misapprehension that the Constitution is a cage and a laundry-list rather than a framework upon which great things have been and still will be built. Many things that are entirely Constitutional are not "in the Constitution," but are allowed under it.