An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor and Wang (1993), to a unique (up to fattening) “flat flow”.