Unless you wanna take apart the entire case and the pump to clean it, the best I can recommend is to drain all 3 plugs underneath the rear diff, the pump, and the trans plug. Let these drain into buckets overnight. Then refill with the 90w mineral oil. The flush before I have never tried as far as using paint thinner or mineral spirits. you can however remove the side cover on the pump housing and rinse with the mineral oil. It is best also to drain this after running the tractor including the pto for about 30 mins so that it will stir up sediments and mix then immediately drain.
After refilling with 4.75 gal of mineral oil. Do not put more than that, otherwise the oil will leak through differential onto your brakes and that'd make it slightly difficult to stop!!! Run the tractor for 25-30 hours and then drain again if it's really really dirty. IF it's milkly, look for a source for water to be penetrating the case (ie. the shifter boot, top cover, side covers, etc.) After the second drain and refill, the oil should be in good working order.