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The flow after the rupture of a dam on an inclined plane of arbitrary slope, and the induced transport of non-cohesive sediment, is analysed using the shallow-water approximation. An asymptotic (analytical) solution is presented for the flow hydrodynamics, and compared with the numerical simulation of the dam-break flood. Differences arise due to the appearance of hydrodynamic instabilities (hereafter called roll-waves) in the numerical solution. These roll-waves point out the unstable behaviour of the dam-break wave. It is found that roll-waves enhance the transport of suspended sediment. The limitations of this simple model to predict the transport of sediment in dam-break floods on steep inclines are discussed. Then, a numerical experiment is designed to analyse the unstable character of the dam-break wave, that constitutes a non-parallel and unsteady base flow. 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 (as it happens in the classical problem of roll-waves over a uniform and steady flow) but also on the non-parallel and time-varying characteristics of the background flow. Consequently, it is also shown that these effects stabilise turbulent roll-waves and raise the critical Froude number required to achieve an unstable flow. This stability result differs with that obtained with a non-parallel spatial stability analysis, pointing out the strong influence of the base-flow time-dependence in the stability criteria. A novel Continuum Mechanics model is presented to study the transport of sediment in a laminar/turbulent free-surface flow. The mixture equations for non-cohesive sediment transport in turbulent free-surface flow are derived from the ensemble averaged Navier-Stokes equations of the three-phases (water, sediment and air). This model avoids the limitations of traditional shallow-water models, and is suitable to study, for instance, the transport of sediment in non-hydrostatic shallow-water flows over bed of arbitrary bottom slopes. The model developed in this work reveals a mathematical equivalence between the propagation of the volumetric concentration of the sediment and the phase function used to capture the free surface. We take advantage of this fact to formulate an explicit Finite Volume Method (FVM) with the exact conservation property, that is implemented in the open source software OpenFOAM. Finally, this model is applied to solve the problem of local scour around a pipeline and the transport of sediment after the rupture of a horizontal dam. It is demonstrated that one-dimensional models based on depth-averaged variables (e.g. generalisations of the one-dimensional Saint-Venant equations to predict mor- phological changes) are superseded by more sophisticated and accurate procedures valid for hyperconcentrated shallow-water flows over bed of arbitrary bottom slopes (e.g. the model described herein).