I have discovered a relatively easy and flexible way to copy all of your Thunderbird email data.
(I'm pretty sure this approach will work for all versions of all operating systems where Thunderbird can be used. I successfully used this technique myself in about October of 2016, with a current version of Windows 10 and a current version of Thunderbird.)
You can copy your data by copying your Profile Folder and then telling Thunderbird where to find your Profile Folder.
Step 1. Find the existing name and location of your Profile Folder by selecting "Help > Troubleshooting Information" in the Thunderbird menu. Then look for "Profile Folder" and click on the Show Folder button. This will display the path of names starting with parent folders and ending with the name of your Profile Folder.
Step 2. Use the facilities of your operating systems (for example, Windows File Explorer) to highlight and then copy your profile folder, with its contents, to a new location (on your new computer, or wherever - with a new profile folder name, if you wish). Be sure you are not copying a profile folder while Thunderbird is running and using that folder. If you have a large amount of email data, the copying process may take a long time.
Step 3. Start the Profile app (on your new computer, or wherever): For example, in Windows 10, a) Close Thunderbird, b) Right-click the Start icon, c) Click Run, d) Enter this in the Open box (without the quote signs): "thunderbird.exe -p" , e) Click OK
Step 4. Use the Profile app to create a new Thunderbird Profile for your new Profile Folder: a) Click the Create Profile button, b) Click Next, c) Enter a name for your new Thunderbird Profile (this does not need to be the same as the name of your Profile Folder), d) Click the Choose Folder button, e) Navigate to and select your new location's Profile Folder that was set up in Step 2, f) Click Select Folder, g) Click Finish
You will now see a Profile app window that displays your new profile in the list of your profiles. (You'll probably have at least one other profile, even if it's just called "default".)
If each time you start Thunderbird you want to select from a choice of profiles, uncheck the "Use the selected profile without asking ..." box.
To automatically use a specific profile, select that profile and check the "Use the selected profile without asking ..." box. (You can always start the Profile app again, to change this setting.)
Close the Profile app by click the Exit button.
The Profile Folder stores everything about your email, including locally stored messages and attachments, address lists, account settings, Thunderbird settings, stored passwords, tasks, calendar data, and data for add-on features.
You can keep a Profile Folder on just about any kind of storage or device. Regularly back up your profile folder. Once you're sure you don't want the data in an old Profile Folder, you can use your operating system to delete it.
Use good judgment about revealing your Thunderbird Profile Folder names and locations to others.