The cooling system is responsible for removing heat from the engine into the environment. Coolant that circulates in the cooling system transfer heat from the engine to the radiator where it is dissipated away with the help of radiator fins and radiator cooling fan.
A transmission transfers power from the engine to the wheels. It is also used to vary the power and torque that is transferred to the wheels.
There are some cars that have transmission heat exchangers attached to the bottom of the radiator.
Mineral deposits as a result of prolonged use of water can go settle at the water passage side of the oil cooler and block it.
You will then have only ATF circulating but not water and hence no heat exchange. This is the beginning of the end of such transmission as the transmission will be overheating especially during high ambient temperatures, sporty or high speed driving.
This is because there would no longer be heat transfer from the transmission fluid to the coolant.
The transmission overheats, ATF gets burned, clutches wear out, solenoids overheat and gets damaged, electrical wires resistances increase and so on.
How to prevent transmission damage due to cooling system problems
This can be prevented by ensuring that the cooling system maintenance is properly carried out; quality coolant should be used, different coolants should not be mixed, cooling system flush should be done periodically, only water being used in the cooling system should be avoided.
We recommend you read this resourceful article on whether it is a bad practice to remove car thermostat.