Whether or not water evaporates, that is the same thing as saying whether or not the air is saturated, depends on both the system temperature and the partial pressure of water vapour in the air. If you reduce the system temperature you move toward saturation. It is also true that if you increase the partial pressure of the water vapour you move toward saturation. And vice versa.
You can change the partial pressure of the water vapour by either changing the amount of water vapour in the air - as by evaporation of water into the atmosphere - or by changing the overall atmospheric pressure - as by going up to a greater height.
So far as we are concerned it is enough to know that when you take a parcel of unsaturated air aloft the cooling efect far outweighs the pressure reduction effect, so the parcel becomes nearer and nearer to being saturated.
I don't know of any CQB questions that deal with this esoteric point, but it is a fact that for any given water vapour content dew point temperatures will be slightly lower at height. Thus the statement that for any given water vapour content dew point remains constant is only true for constant atmospheric pressure - usually we mean sea level.
Dick W