The current package of firefox-esr in stretch (68.3.0esr-1~deb9u1) reports a
bigger version number than the current package in buster (68.3.0esr-1~deb10u1)
due to the inclusion of the build timestamp. The version reported by stretch is
68.3.0_20191206235801/20191206235801 and buster
68.3.0_20191203235607/20191203235607.
This information is then used by firefox to fill the compatibility.ini in the
profile and after an upgrade from stretch to buster leads to firefox prompting
to either quit or create a new profile because it detects that an older version
of firefox is trying to open a profile last used by a newer version. I believe
using the same profile should be safe, as the underlying upstream version is
the same at the moment and should probably never be newer in oldstable than
stable.
As a workaround it's possible to start firefox with the parameter "--allow-
downgrade" or manually delete the line starting with "LastVersion=" in the
compatibility.ini in the profile but less savy users may use access to the data
in their old profiles (the data is still there but a regular user will probably
not be able to access it without further instructions).