We study the structure and propagation of combustion waves in thermite systems in which the heterogeneous mixture of solid reactants initially forms a solid matrix. This solid matrix is destroyed by the combustion wave as one of the reactants melts, producing a bath of one reactant in which particles or droplets of the other reactant are suspended. Gravity can lead to relative motion and hence separation of the reactants. We formulate a simplified mathematical model of this process in order to gain insight into how relative motion and separation affect the characteristics of the combustion wave. We analytically determine the structure of one dimensional uniformly propagating combustion waves and show that depending on the characteristics of the system, such solutions may fail to exist or may be nonunique. We also investigate the dependence of the combustion velocity on gravity.