We prove the property of finite speed of propagation for degenerate parabolic equations of order 2m [ges] 2, when the nonlinearity is of general type, and not necessarily a power function. We also give estimates of the growth in time of the interface bounding the support of the solution. In the case of the thin-film equation, with non-power nonlinearity, we obtain sharp results, in the range of nonlinearities we consider. Our optimality result seems to be new even in the case of power nonlinearities with general initial data. In the case of the Cauchy problem for degenerate equations with general m, our main assumption is a suitable integrability Dini condition to be satisfied by the nonlinearity itself. Our results generalize Bernis' estimates for higher-order equations with power structures. In the case of second-order equations we also prove L[infin] estimates of solutions.