Setting up a new machine and wanting to move my Thuderbird profile to the new machine. Having setup the previous profile in a custom location under my $HOME directory makes this relatively simple. The trick is to find the profile.ini file used by Thunderbird and configure it on the new machine to point to the profile location.
On the mac look inside ~/Library/Thunderbird/.
There edit the profile.ini file.
Here is the before and after for me.
[Profile0]
Name=default
IsRelative=1
Path=Profiles/0uqtzy4a.default
Default=1
[Profile0]
Name=default
IsRelative=0
Path=/Users/brad/mail
Default=1