Gravity–driven Stokes flow of an ice layer sitting above a layer of ice mixed with sediments down an inclined plane is considered. The ice in the top layer and the ice and sediment constituents in the bottom layer are treated as density–preserving non–Newtonian fluids and their interface is treated as fixed. The constitutive relations of the constituent stresses and interaction force (between ice and sediment) are additively composed of thermodynamic equilibrium and non–equilibilium contributions of which the former are given by a thermodynamic potential, the mixture inner free energy, whose form is motivated by the isotropic finite strain theory. In the thin–layer approximation, the lowest order governing equations emerge as the limiting equations, valid when the shallowness parameter becomes infinitely small. In this context the non–dimensionalized momentum balances allow determination of the across–flow variations of the volume fraction of the sediments and mixture pressure, independent of those of the streamwise constituent velocities. The qualitative agreement of these profiles with corresponding information from observations at the basal layer of glaciers and ice sheets gives reason for optimism of such a theoretical formulation.