How a poorly maintained cooling system can damage a car's transmission
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, oil gets burned, clutches wear out, solenoids overheat and gets damaged, electrical wires resistances increase and so on.
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.
You can refer to the links below for more on cooling system maintenance.