We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in IRN. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat phi-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat phi-curvature flow starting from a compact convex set is unique.