Proper ATF replacement procedure is quite simple actually.
So here it is:

Raise you vehicle slightly. You need decent size vessel to fit under.
Disconnect ATF return line at cooler (line #2 on diagram).
This diagram applies to Gen 2 Montero only but others (Sport, Gen 1, Gen 3) should be very similar.

Direct it into your catch container via funnel like I did or attach longer hose in it's place. Even clear vinyl tubing out of Home Depot will do.

Shift your AT into neutral. Make sure your drain fixtures are not going to fly around. Turn on your engine and let idle for 30 sec to 1 min (NOT MORE). This will pup out whatever fluid can be picked up from AT sump out into that container.
Shut off engine.
Drain remaining ATF from transmission sump vi drain plug.

Install the drain plug back and tighten it. Do not do anything with container upfront yet.

Now there is still a little fluid left in transmission internals.
To pump that out add maybe 4-5 quarts of NEW ATF to transmission vi measuring stick tube. Don't worry about overfilling yet.


Turn on the engine again (transmission still in Neutral(N) not P. This will pump out rest of old ATF and replace it with new fluid. Some of new fluid inevitably will be lost but that's the price.

Remove that catch container and put everything upfront back together. Add ATF to required level. Don't forget that you have check fluid level with running engine and transmission in Neutral not P. Enjoy smooth shifting transmission once again.

Now on that screen filter. IMHO there is no good reason to replace (I assure you dealer won't do anything about it) or wash it unless there was a problem with transmission fluid (burned smell, weird color etc) or your transmission overheated etc.

This procedure has pretty much exactly effect (and works) as dealer's mythical ATF changing machine. And no, they don't use and detergent, I personally wouldn't want them to anyway. If there is a problem requiring special cleaning washing it probably means you have something bigger to worry about. Properly functioning hydraulic systems are always very clean inside.