We compare two models describing the dynamics of phase separation of incompressible mixtures: A local model proposed by de Gennes, and a nonlocal model proposed by E and Palffy-Muhoray. We find that in the interfacial regime, the local model gives rise to interfacial motion via the Mullins-Sekerka law for moderate quenches, and surface diffusion for deep quenches. The interface dynamics is slowed down sharply as the quench depth is increased. The nonlocal model, on the other hand, has an additional convective mechanism which gives rise to motion by Hele-Shaw dynamics. This additional mechanism is insensitive to the quench depth. While both models explain qualitatively the observed pinning phenomenon for deeply quenched off-critical mixtures, only the nonlocal model predicts the correct dependence of the coarsening rate on the quench depth. (C) 1997 American Institute of Physics. [S0021-9606(97)50644-1].