The wet-bulb measurement is independent of airflow above a certain point. A figure of 500 ft/min is often cited as the minimum.
Water evaporating from the wet-bulb cools it. As it cools, heat transfer occurs from the air to the bulb. The rate of evaporation slows as the bulb cools. Simultaneously, the rate of heat transfer increases as the bulb cools (more heat moves from the surrounding air to the bulb). At a certain temperature these two factors balance and the temperature becomes steady. At the wet-bulb temperature, the energy reaching the bulb by convection just equals the energy carried away by the evaporating water molecules.
The two factors (heat transfer and evaporation) are both controlled by the insulating effect of an air layer at the surface of the bulb. When airflow increases or decreases, the layer is affected and both factors are similarly affected. For this reason, the wet-bulb temperature is not affected by airflow (above a certain point). Heat transfer by the moving air is called convection.
High airflow is desirable to minimize the impact of things (radiation and conduction) that affect amount of heat or energy reaching the bulb. For example, the metal in an rtd conducts (conduction as opposed to convection) some energy from the rtd mount to the tip. Also, heating coils in view of the bulb can radiate (radiation instead of convection) energy to the bulb. Hence, dry- and wet-bulbs may have heat shield in some locations.
Below a certain air velocity, the mode of convection changes from turbulent to laminar. This depends on the bulb diameter and other factors. This is why most books say 500 ft/min and up is ok.