The definition of the thermodynamic wetbulb is the temperature of adiabatic saturation. i.e. it is the outlet temperature of air sent through a 100% effective adiabatic cooler.

Evaporative coolers can be modeled as adiabatic coolers with a prescribed effectiveness. With minimal piping and pump heat gain, the recirculating fluid can only exchange heat with the air stream. Without a heat source/sink, the fluid will quickly come to equilibrium with the inlet air, and no net heat transfer occurs. If the inlet air is not saturated, there will be sensible and latent heat transfer, but two will always cancel each other out. By the definition of the thermodynamic wetbulb, you should expect the wetbulb to be nearly constant in any evaporative cooling model.