when air is rising or descending, what happens to its temperature?

Answer 1

Some chicken-egg here.

Air that is warmer than surrounding air tends to have more molecular motion, and so more volume per mole of atmosphere. Thus, it is less dense, and by bouyancy, tends to rise. The opposite is true for air that is cooler than surrounding air.

Now, as that warmer air rises, it tends to lose heat (by entropy) to its cooler surroundings until it reaches equilibrium (equal temperature) and stops rising. Similarly, as cool air falls, it tends to gain heat from its warmer surroundings OR it hits "bottom" and comes to equilibrium.

In weather systems it is much more complicated, since volumes of air displace other volumes of air, and convection currents can be set up where a heat-driven conveyor moves warm air and cold air masses in large systems.

Answer 2

The temperature of rising air declines due to the decrease in pressure. This is know as the adiabatic lapse rate and takes place even if there is no net transfer of heat to the surrounding atmosphere. Conversely, air that sinks warms up.