Field measurements suggest that the relationship between velocity and discharge in steep rough upland channels may be nonlinear at low‐flow stages but approximately linear at high‐flow stages. This paper describes a generalized kinematic channel network routing model which can allow a flow relationship that describes both high‐ and low‐flow characteristics. The model is applied to the channel network of the Institute of Hydrology's River Severn experimental catchment and, for this specific application, proves to be equivalent to a linear model based on a constant kinematic flow velocity. A strategy for choosing and applying a flow routing model for use in small catchments is suggested.