Making a backup with Mailbird is easy!
All you need to do is open your file explorer, and go to C:\Users\"your user name"\AppData\Local and copy the Mailbird folder that is there. The Mailbird application should be CLOSED at the time of copying.
Next, Install Mailbird in your new PC. After the installation is complete, go to the same location as the above, on your new PC and paste/overwrite the existing Mailbird folder with the back up that you've created from your old one.
And that's it! You can open and use Mailbird on your new PC with all your existing settings and email.
Some of the directories mentioned above might be hidden by default. In the "folder options" of Windows Explorer there is a setting to show hidden folders - this needs to be enabled.
For Windows10, at Windows Explorer, click the "View" tab, enable the "Hidden Items".
If you need to move emails from one server to another, then, unfortunately, there's not much you can do in Mailbird. You can ask your email provider if they provide any kind of a backup tool, or use a 3rd party service like:
Google Takeout for Gmail
Also, if you are using Gmail, there are a number of Chrome extensions (like the aptly named Save Emails to PDF) that allow you to export, download, or print your messages as PDFs, including multiple emails at a time.
Also, you can take a look at Mailstore: https://www.mailstore.com/en/products/mailstore-home/