The first question that needs answering is -- are you using POP3 mail or WEB mail?
If you are using POP3 mail, then it will need to be moved - because, it is actually on your local computer.
If you are using WEB mail, then it does not matter - because, your mail is on a web server somewhere.
If you are using Outlook, then I imagine it is downloading your mail to your local computer.
If you were using WEB mail, then you would be accessing it through a browser.
So - yes - you will probably need to export your mail [ to a single file if possible ] - when the time comes - to move your mail to the new computer.
Then you will copy / move that file to the new computer and import it into the mail tool ( Outlook, etc. ) that is on the new computer.