In a situation where the temperature is above freezing and snow begins to fall, the 2m above ground temperature will often quickly drop to 32 F and remain at this temperature. As snow falls into a layer of above freezing air, the snow can cool the air in three ways: sublimation, melting, and evaporation. The most significant cooling occurs from the evaporation process as snow falls through above freezing air. Melting also cools the air somewhat. The evaporation process absorbs about 7.5 times as much latent heat as the melting process. Therefore, snow falling through dry air will cool the air much more quickly than when snow falls through saturated air. The following essay explains the rate snowflakes melts as they falls in detail:

As heavy wet snow falls, the air cools from evaporation, melting and to a certain extent by cooler air brought down to the surface from aloft (conduction into colder precipitation particles).

When the 2m above ground temperature is above freezing (38 F for example) and moderate to heavy snow begins to fall, the temperature will tend to quickly fall to 32 F and then stop falling. Why does it tend to not fall much below 32? In the absence of significant CAA, the tendency for the temperature to drop to 32 F and then stop falling is very common when wet snow falls into air that is several degrees above freezing. The reason this occurs is due to the release of latent heat. Wet snow has a high water content. Also, some of the wet snow will melt as it hits the ground (especially if the ground surface temperature is above freezing). In order to freeze the water content in the snow and the water on the ground, the temperatures would need to fall to freezing and then below. The freezing process RELEASES LATENT HEAT. Since the release of latent heat is a warming process, the temperature will tend to stay around freezing when there is an ice / water mixture on the ground and falling from the sky. Ice water will remain at 32 F until all the ice has melted or all the ice has frozen. Since wet snow represents water in phases of a liquid and a solid, the temperature will quickly cool to 32 F and remain near that temperature throughout the wet snow event. This is especially evident when no significant thermal advection and no excessive wet-bulb cooling occurs to contaminate the process.