Pumping of charge in phase-coherent mesoscopic systems due to the out-of-phase modulation of two parameters has recently found considerable interest. We investigate the effect of inelastic processes on the adiabatically pumped current through a two terminal mesoscopic sample. We find that the loss of coherence does not suppress the pumped charge but rather an additional physical mechanism for an incoherent pump effect comes into play. In a fully phase incoherent system the pump effect is similar to a rectification effect.