How are mixing valves installed? Do they affect just the hot water line?
The purpose of these thermostatic (static: constant, thermo: temperature) mixing values is to give you a controlled-temperature output even when the hot and cold input temperatures can vary.

When you take a shower, the temperature can fluctuate because of the cold water (water from pipes in the walls vs water that just came in from underground) and because the water heater is "running out" of water and getting cooler.  The value works by mixing cold water in with the hot to give a constant output.

So, let's say you set the valve for 120, and have hot water at 140 and cold water at 60 degrees.  I'm going to make up some math on the fly and say that the valve will mix about 80% hot water with 20% cold to cool it off down to 120.  As the incoming temperature drops, it uses less cold.  So when the water heater gets down to 125, it's going to be more like 95% hot and 5% cold.  Finally when the heater is down to 110 degrees, you're going to get full hot water and then you're left on your own to adjust the shower handle.

The speed at which the adjustments happen will be device dependent an "adding hot water to the cold" or "adding cold to the hot" is really just two ways of saying the same thing.  Also I believe that some models only have one "valve" to change the amount of hot or cold and others have two separate valves (maybe for greater temperature swings or faster action?).