The unstable nature of kinematic waves, resulting from outburst floods on steep inclines, is investigated theoretically using the shallow water equations. A numerical experiment is designed in order to efficiently perform a stability analysis. By analysing the linear and non-linear numerical evolution of small perturbations, it is possible to reveal how the nature of the ensuing flow depends not only on the Froude number but also on non-parallel and time-varying characteristics of the background flow. It is also demonstrated that these effects stabilize turbulent roll-waves and raise the critical Froude number required for instability.