Exporting to Thunderbird

Is a migration process required?

A key question is whether you need to manually migrate anything at all. Please note the following:

If your account is using IMAP, then you shouldn't need to migrate since Thunderbird will download all of your folders and mail for you.
If your account is using POP or POP3, then Thunderbird can download messages from your Inbox. Messages in other folders will need to be manually migrated to Thunderbird.
Messages stored on your local machine will need to be manually migrated to Thunderbird.

Within Postbox, compact your folders by selecting Compact Folders from the File menu. If your mailboxes are quite large, this could take several minutes. Quit Postbox.
As a precaution, backup Thunderbird by making a copy of your profile directory. Information on how to do this can be found within this article. Be sure to make a backup of your Postbox profile as well.
Find the mail folders you would like to copy from Postbox. On the Mac, they are located at: [your home folder]/Library/Application Support/Postbox/Profiles/[your profile]/ and on Windows at: C:\Documents and Settings\[Windows user name]\Application Data\Postbox\Profiles\[your profile].
Mail is contained in the following folders: Mail (for POP mail and Local Folders) and ImapMail (for IMAP mail).
Copy the contents of your Postbox ImapMail folder to your Thunderbird ImapMail folder located on the Mac at: [your home folder]/Library/Thunderbird/Profiles/[your profile]/, and on Windows at: C:\Documents and Settings\[Windows user name]\Application Data\Thunderbird\Profiles\[your profile].
Repeat this process for Mail/Local Folders, and Mail.
Start Thunderbird.

For more migration options, please see this article.