We formulate and analyze a model describing the combustion of porous condensed materials in which a reactant melts and spreads through the pores of the sample. Thus there is liquid motion relative to the porous solid matrix. Our model describes the cases when the melt either fills all the pores or when some gas remains in the pores. In each case the melt occupies a prescribed volume fraction of the mixture. We employ both analytical and numerical methods to find uniformly propagating combustion waves, to analyze their stability and to determine behavior in the instability region.