A diffusion wave model of distributed catchment dynamics is presented. The effects of catchment topography and river network structure on storm-flow response are incorporated by routing surface runoff in cascade throughout a digital elevation model (DEM) based conceptual transport network, where the Muskingum-Cunge scheme with variable parameters is used to describe surface runoff dynamics. Dynamic scaling of hydraulic geometry is also incorporated in the model by using the 1953 “at-a-station” and “downstream” relationships by Leopold and Maddock. Numerical experiments indicate that the model is more than 98% mass conservative for possible slope and roughness configurations, which may occur for hillslopes in a natural catchment. Fluctuations in the simulated discharge may occur in response to discontinuities in rainfall excess representation if Courant number Cu during the simulation exceeds a threshold of about 3. Catchment scale simulations with different temporal resolution show that the model response is independent of structural parameters (model consistency). Also, the overall accuracy is preserved for computationally inexpensive space-time discretizations (for which Cu > 3) because fluctuations that may occur at the local scale are dampened when propagating downstream. Comparison of model results with observed outlet hydrographs of the Rio Missiaga experimental catchment (Eastern Italian Alps) show this approach to be capable of describing both overland and channel phases of surface runoff in mountainous catchments.