Coolant can gain access to transmission through the transmission oil cooler. The oil cooler can leak internally causing the coolant and oil chambers to be linked.
Coolant getting into the engine lubrication system can cause damage to the engine. It is mostly caused by bad head gasket, bad radiator that has oil cooler integrated or a bad engine oil cooler.
There are salient causes of overheating that are hard to troubleshoot. One symptom is coolant bubbling even before engine attains operating temperature.
Heat generated from automatic transmissions are transferred to the cooling system for dissipation to the environment through the transmission oil cooler.