In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients.The main unknowns of the system are the sediment thickness h, the L surface concentrations $c_i^s$ in lithology i of the sediments at the top of the basin, and the L concentrations c i in lithology i of the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for $c_i^s$ with a linear advection equation for c i for which $c_i^s$ appears as an input boundary condition. For this coupled system, a weak formulation is introduced which is shown to have a unique solution. An implicit finite volume scheme is derived for which we show stability estimates and the convergence to the weak solution of the problem.