  We provide a new convergence proof of the celebrated Merriman-Bence-Osher
scheme for multiphase mean curvature flow. Our proof applies to the new variant
incorporating a general class of surface tensions and mobilities, including
typical choices for modeling grain growth. The basis of the proof are the
minimizing movements interpretation of Esedo\u{g}lu and Otto and De Giorgi's
general theory of gradient flows. Under a typical energy convergence assumption
we show that the limit satisfies a sharp energy-dissipation relation.
