What happens to air when it sinks?
It warms and the relative humidity drops, and the skies clear.
As an air parcel sinks the pressure of the surround air around it increases. We know from Gay-Lussac's law that pressure and temperature are proportional. Therefore as the pressure increases the temperature increase.
Since there is no addition of water to a sinking parcel of air, meaning the dew point is going to remain constant, the relative humidity will drop. For example, if the air is 10 degrees and the dew point is 10 degrees you have 100 percent humidity and cloud forms. As the air sinks the temperature might warms to say 20 degrees but without addition of water the dew point will still be 10 degrees. The relative humidity will be about 75% and no cloud will form. As the air continues to sink it will continue to dry out.