  We consider nonlocal curvature functionals associated with positive
interaction kernels, and we show that local anisotropic mean curvature
functionals can be retrieved in a blow-up limit from them. As a consequence, we
prove that the viscosity solutions to the rescaled nonlocal geometric flows
locally uniformly converge to the viscosity solution to the anisotropic mean
curvature motion. The result is achieved by combining a compactness argument
and a set-theoretic approach related to the theory of De Giorgi's barriers for
evolution equations.
