Best is to Use IMAP if you insist on POP se the alternative at the bottom.
Since you are using the POP protocol in your mail client. Mails are most probably being downloaded to your email client and removed form the server
(check your email client settings to verify this)
If you were running IMAP already you would only need to to Step NR 3
1) Set up IMAP on your old mail client.
Disable checking of email with the POP account but do not remove it
and re add another the same email account in your windows email client, but this time with IMAP support instead of POP support.
Since IMAP will not download and remove emails from the mail server ,but simply make an cached version of your emails in your email client. Allowing you to read delete and otherwise interact with your emails from multiple devies.
2) Move emails back to server
Once that is done , drag and drop all your emails from your pop3 inbox in your mail client into the IMAP inbox you should be having.
This will basically re-upload the emails onto your e-mail providers server.
3) Configure IMAP on thunderbird.
Once all email have been moved back.
You can simply configure your thunderbird on linux or any email client
to gain access to your emails.
In general IMAP is always preferred over POP as it has better support for handling email when using multiple email clients in multiple locations.
If you insist on continuing using POP:
Since the Thunderbird file format is the same between different operating systems you can actually extract the data file manually from the install folders.
But it might be easier just to install the import export add on ,
export the emails with it , move that file to the new place and import it with the same add on.