I'm not sure what you mean. What I try to achieve is start a new firefox window (for the normal user) from a root terminal. So part of the command has to be "sudo -u normaluser" or "su -l normaluser" or something.
sudo has a --preserve-env=list option but you must know the value for DBUS_SESSION_BUS_ADDRESS. That's usually
unix:path=/run/user/1000/bus
To be 100% sure of the value, as you are root you could look into the environment of one of the running Firefox processes. Or wrap Firefox in a script that echoes that variable to a file before starting the browser.
DBUS_SESSION_BUS_ADDRESS=unix:path=/run/user/1000/bus \
DISPLAY=:1 \
sudo -H -u normaluser \
--preserve-env=DBUS_SESSION_BUS_ADDRESS,DISPLAY \
firefox
I assumed that your DISPLAY is :1, which is what my Debian has set for me. That's another variable that you could read from the environment of Firefox.
You might have to pass/preserve other environment variables to make Firefox work.
(You could also test the extreme case: Just copy all the env vars.)