We study the separation in the binary and ternary mixtures of the water/surfactant C12E5/polymer PEG system. The phase separation in the mixtures at late stages is governed by two distinct mechanisms: the coalescence-induced coalescence and the droplet evaporation mechanism. We show that when the coalescence-induced coalescence process is globally terminated in the sample consisting of a dense system of domains, another mechanism, which we call the collective droplet evaporation, starts to dominate. It manifests itself as a front of "evaporating" domains, which propagates at constant speed in the system. We show that the collective evaporation is induced by the gravitational drift of large droplets.