I don't know if there's consistency across brands in that regard or not. I know Mopars generally routed the fluid to the cooler and then to the lubrication points throughout the gear train and bearings, so the pressure through the cooler is on the order of 30-40 PSI. If there are automatics that return cooled fluid directly to the pan instead, the pressure would be significantly lower.
So in this case, the fluid pressure is generally higher than the cooling system, but its not a fail-safe. The instant you turn the engine off the trans fluid pressure drops to zero, but the cooling system pressure can continue to rise. The one time I had a trans cooler fail, things went both ways. There was brown mud in the radiator from oil pumping into the cooling system when the engine was running, and there was brown mud in the transmission pan from coolant getting pushed into the cooler when the engine shut off. Oh, and a quirk of chemistry is that transmission fluid makes the silicates precipitate out of old-style coolants. So it really IS brown mud... sandy mud.