Changed out thermostat on 02 mitshibu montero limited.hot/cold gauge still moving into hot and drop a little and back into hot zone.

it could be you have an air lock in the cooling system
remove the rad cap from the rad, start the engine and let it run until the coolant drops (add to fill it back up) or until the coolant boils out of the rad spout.

maybe pull the heat sensor. -replace it OR clean it off, reinstall it.

Faulty coolant temperature sensor and a faulty heater control valve.

ADD COOLANT IN THE COOLANT OVERFLOW JUG UNTIL COOLANT LEVEL STOP DROPPING AND COOLANT LEVEL STAY AT THE FULL COLD MARK.PLACE COOLANT OVERFLOW JUG CAP BACK ON JUG.CRANK VECHICLE UNTIL ENGINE START WARMING UP. STOP ENGINE WAIT FOR A WHILE UNTIL COOLANT COOL DOWN TO KEEP FROM GETTING SCALDED. USE A LARGE RAG OVER THE OVERFLOW JUG CAP OPEN OVERFLOW JUG CAP A LITTLE AT A TIME REMOVE OVERJUG CAP CHECK COOLANT LEVEL ADD MORE. COOLANT IN THE OVERFLOW JUG.THEN START VECHICLE AGAIN.LET ENGINE IDLE UNTIL TOP RADIATOR GET HOT.THEN THERMOSTAT WILL BE OPEN.WHILE ENGINE IDLING WATCH THE TEMPERATURE GAUGE IN VECHICLE IF TEMPERATURE GAUGE START CLIMBING IN THE HOT ZONE.TURN OFF VECHICLE LET IT SET A WHILE THEN USE LARGE RAG REMOVE OVERFLOW JUG CAP.ADD MORE COOLANT, PLACE CAP BACK ON START VECHICLE AGAIN KEEP DOING THIS PROCEDURE UNTIL TEMPERATURE STOP CLIMBING HOT ZONE.AND WHEN COOLANT LEVEL STOP DROPPING IN THE COOLANT OVERFLOW JUG.AIR IS OUT THE SYSTEM, BUT KEEP COOLANT LEVEL IN THE COOLANT OVERFLOW JUG AT THE FULL COLD MARK.DONT OVERFILL.DONT LET COOLANT OVERFLOW JUG RUN DRY IF SO AIR WILL ENTER COOLANT SYSTEM BECAUSE COOLANT LEVEL WILL BE LOW ENGINE WILL START OVERHEATING.

Probably a stuck open thermostat or the thermostat is the wrong temp range for your vehicle.

mightbe defective thermostat

The thermostat may be wearing out. A thermostat controls the engine temperature to keep it at a consistent temperature, The other fan will probably turn on if you turn on your Air Conditioner.

2003 Mitsubishi Montero Limited. 97k miles. Temp gauge is erratic when actual temp is normal. How do I fix this?

Follow the top radiator hose down to the water outlet. The thermostat is inside the water outlet,Just remove the bolts,replace thermostat,replace gasket,bolt back in place.

It sounds like your thermostat is broken. The metal retaining tabs that hold the thermostat together can break over time. I would recommend pulling it out and checking it. When installing a thermostat there is an arrow on the edge. Make sure it points up. toward the top of the cylinder head.

As long as your gauge is still moving up, there's no real problem. You have heat coming from the heater. It's only a problem if it's getting too hot. What's going on is that you don't let the vehicle run long enough, and it's so cold outside that the engine stays cold. It raises in temp when your stopped because there's no airflow through the radiator. The thermostat just heats up the engine block, and won't change the reading on the gauge. When this goes bad your car will overheat in just a few miles.