So you have a new system and on the old system you have Mozilla Thunderbird as your email. Recently this was my situation. I just built a new Kubuntu 16.04.1 LTS system and wanted to migrate from my aging Macbook, amazingly it is very simple to do.
First step is to shutdown Thunderbird on the old system and copy the profile folder. To a USB stick or a shared location.
On Mac OSX
Windows 7 and above
C:\Users\<Windows user name>\AppData\Roaming\Thunderbird\Profiles
On your target system open up Thunderbird and create your email account, moving your profile will not do this portion.
Next copy or move the profile folder to the desired location on your new system in the correct profile folder.
Open up the profiles.ini file in a text editor. The file is located in the application data folder for Thunderbird.
In the profiles.ini file, locate the entry for the profile you’ve just moved. Change the Path= line to the new location.
Save profiles.ini and restart Thunderbird and Addresses and messages will all be visible on your new system