Thunderbird can only import mail from Outlook Express if the two programs are on the same computer. This creates something of a catch 22 for those that have backed up their DBX files from Outlook Express, and no longer have access to the program.
In the past few days, I was lurking on a Microsfot Support Forum, and mention of theUnDBX utility came up. Now the internet is full of paid utilities that will read DBX files and convert them, but really who wants to pay for a program that they will probably use only the once. This one is free, and released under the GNU GLP V3 open source license.
How does this help in getting DBX mails into Thunderbird, it converts them to the industry standard EML file format. The addon Import/Export Tools can import EML files into Thunderbird.
So the round trip is use the UnDBX tool to convert your mail into EML and then Use the Import Export tools to import it into Thunderbird.
I think I have finally figured out how to do the WAB to Thunderbird part of the process. A small largely undocumented program called Dawn. I have linked to the download site that does contain some instructions for use, in my own case I had to unzip the package and double click on the MSI file. The setup program did not appear to do anything. I would love to hear from anyone how Dawn went for them. Other than the quirk on the setup it worked flawlessly for me.
Addendum 17 March 2013
Philip Sargent has advised me that under Windows 7 Dawn needs to run in XP compatibility mode.
Addendum 30 January 2016
I think these days using the windows contact to import the WAB file and then instructing Thunderbird to use that is a better way to go with address books. See my Devember 2013 post on the Windows address book here