We compare solutions of a class of degenerate parabolic equations on a Riemannian manifold $M$ with solutions of the equation on a model manifold. The class of equations under consideration contains both the parabolic $p$-Laplace equation and the porous medium equation. We prove that, under curvature conditions, solutions on model manifolds induce sub- or supersolutions on $M$. Using this result, we obtain curvature-dependent estimates for the speed of propagation of solutions.