We study combustion waves propagating through a heterogeneous powder mixture which initially forms a hard porous matrix. The matrix is destroyed by the propagating combustion wave, due, e.g. to melting of some of the components of the mixture. Thus, a liquid bath is formed which contains a suspension of particles. The burning of the sample occurs in a gravitational field. If the gravitational field is sufficiently strong, and the densities of the species sufficiently different, then there is relative motion of the components which affects the composition of the species in the combustion wave, its structure, and therefore, the stability of the propagating wave. Depending on the direction of propagation of the wave and the relation between the densities of the species, two different processes can occur. One is the separation process which occurs either when the combustion wave propagates downward (i.e. in the direction of gravity) and the initial reactant is heavier than the product, or when the combustion wave propagates upward and the initial reactant is lighter than the product. The other is the mixing process which occurs in the opposite situations. We find the characteristics of the uniformly propagating combustion waves in such systems and show that generically the mixing process stabilizes, while the separation process destabilizes the propagation of the combustionwave.