The first thing I would do is try to contact the provider, and see if they can help export it, if not give you the POP3 or IMAP settings. In some countries offering this service may even be required by law.
If that yields no results, I have a few options in my RoundCube webmail, which I hope you have too.
One is to highlight an individual mail, click the More button and Select Download (.eml). Unfortunately that only appears to work on individual e-mails.
Another is simply to forward each e-mail individually to your new account or some temporary address/webserver.
But actually it seems there's a better way:
If you highlight multiple files and select Forward (or click the arrow and select Forward as attachment) it appears as if what you're sending is an e-mail with all the e-mails selected attached as .eml files. Maybe you can try that selecting all of them, and see if you can import them successfully afterwards? How that works if the e-mails themselves have attachments I'm not so sure about, but at least this could mass export all the ones without attachment if not all of them in general.
These .eml files can probably be imported into your new host, or at least imported to a mail program, that can then in turn move it to the new host if you have IMAP access to that.
Some information on importing .eml files into the mail client Thunderbird, as an example: http://www.emailadept.com/import-emails-to-thunderbird.htm