Yeah, that's called "flushing".

Reverse the flow and its "backflushing"

I wouldn't try to clean the heater core with cleaners, as that's had remarkably BAD results for me in the past.

One thing you could do to eliminate the heater core as a potential suspect would be to take it out of the equation, by running a hose between the inlet/outlets to it, "bypass" it, I guess.
I'll say one thing about the "mood swings" of the temp needle, if you're working a truck that does this, it does NOT seem to do it under heavy towing conditions.
As far as the needle's actual tendency to move, I'm convinced that having the sending unit clear on the other end of the head from the thermostat, also plays a role in these swings.
Put the sender up by the t-stat and you'll likely have a more consistent and accurate reading.
Mark.