Your engine and transmission are always under more stress when towing or even if you just have a heavy load in the vehicle. More weight causes harder work which produces heat. Heat is your enemy.
Now, more specifically to your question. It is better to have the engine running at a constant RPM that is about 2/3-3/4 of the engines maximum RPM level rather than to have the trans hunting for gears. Some people are afraid that engine uses more fuel at higher RPMs. That isn't necessarily true. If the engine is lugging and hunting for gears, it is probably using more fuel than if you leave it in a lower gear and run the engine at a constant RPM while pulling up a grade (even a small grade).
The higher the RPM the more HP the engine develops and the engine has to work less when it is at a higher RPM (as long as you aren't pedal to the metal) and it produces less heat.
When the trans is "Hunting" for gears, that puts a lot of stress on the (transmission) torque converter and develops a lot of heat. Most of that heat is dumped into the vehicles radiator which in turn will make the engine run hotter. So the answer is....Heat is hard on both engine and trans. They are both connected by the same cooling system and a hot trans = a hot engine and visa versa. That's why it is very important to have the largest auxiliary trans cooler that you can fit in the vehicle.
The other killer is pulling a hill at low speeds (under 35) or sitting in traffic. There is not enough airflow through the cooling system to remove the heat fast enough. That is why I always recommend a trans cooler with a fan.
Here's an example of the fan assisted trans cooler on my Silverado V8. I have the fan wired to the engine cooling fan with a relay so that every time that the radiator fan comes on, the trans cooler fan comes on at the same time.
Another tip on "Keeping Cool" is to use synthetic fluids. Synthetic fluids have far better cooling properties than petroleum based fluids. They also have a higher maximum operating temperature....it's all a win-win. Synthetics are more expensive but it has to be changed far less frequently which makes the fluid changes cost no more but the product is far superior.
