I know all about the OLD wisdom which was about draining an engine with the oil hot back when everyone used single weight fluids.
Personally I look at it another way and that is by draining the oil before running the engine, the oil has already drained down to the lowest point as well as any contaminants. Since I am draining at the lowest point, I figure I get more of the oil and contaminants out by draining it after the engine has sat for some time.
The same goes for the transmission.
Besides the obvious that it is easier to work on a cold engine, I can see no advantage to warming the engine and transmission up before draining the old fluid. I've been doing it this way for years on my motorcycles, airplanes, trucks, RV's, racing and street vehicles.