The temporal evolution of nonequilibrium coverage profiles in the Pb wetting layer on the Si(111) surface is studied using low energy electron microscopy. The initial coverage step profile propagates rapidly at a constant velocity with an unperturbed shape. A model is proposed that attributes this nonclassical equilibration behavior to the diffusion of thermally generated adatoms on top of the wetting layer. This model can account for the observed convectionlike mass transport, as well as its dramatic dependence on Pb coverage. Such anomalous mass transport is believed to facilitate the remarkably efficient self-organization of uniform Pb quantum island height on the Si(111) surface that was observed previously.