remote display does not start a NEW client

I routinely ssh to another system on my local network and run
mozilla/firefox from that remote system to display to the desktop I am
currently sitting in front of. If the desktop I am currently sitting in
front of already has another browser window of the same browser running
(i.e. mozilla - local and mozilla - remote  or  firefox - local and
firefox - remote) instead of starting a new master browser window from
that remote system, I get another child window from the local system. If
instead I have non-matching browsers running locally and remotely (firefox
- local and mozilla - remote for instance) I do get a new master browser
window from that remote system.

The later is the behavior I am looking for. If I am running a browser
window remotely it is because I want to access some bookmark or history
that is only in the profile of the remote user or I want to open a file
from that remote filesystem that is not directly web available from my
local system.

Is there some way I can use some sort of command line switch or something
to prevent this child local window thing from happening so when I do this
I always get a paprent master browser window from the remote system?
__
sad