We consider heterogeneous combustion in a porous medium subject to gravity-induced buoyant forces. A vertical sample, open to Row at the top and bottom, is ignited at the top. Buoyancy causes the hot gases to leave the sample through the top, thus drawing in fresh cool gas, containing both oxidizer and inert gases, through the bottom. The incoming gas supplies the reaction with oxidizer. In contrast to forced filtration, in which the flux of gas into the sample is prescribed, here the incoming flux is determined by the combustion process itself. This configuration describes smoldering, self-propagating high-temperature synthesis (SHS) of advanced materials, and a host of other applications. Combustion waves are described for both adiabatic and nonadiabatic conditions in which heat is lost to the external environment through the sides of the sample. The heat loss causes the temperature in the product to decay from the combustion temperature to the temperature of the external environment in a region termed the cooling region. Two nonadiabatic cases are considered, distinguished by whether or not the sample is sufficiently long and the heat losses sufficiently large that the cooling region behind the reaction site is completely contained within the sample. When it is totally contained in the sample, we describe traveling wave solutions whose shape does not change in time, provided there is no net production of gas in the reaction. When it is not completely contained within the sample, we describe quasi-steady combustion waves which change slowly in time due to the increasing buoyant flux as the combustion wave penetrates farther and farther into the sample. Solutions are categorized as gas deficient when the oxidizer is completely consumed, solid deficient when the solid fuel is completely consumed, or stoichiometric when both oxidizer and solid fuel are completely consumed. Extinction is found to occur for solid deficient and stoichiometric solutions when the buoyant flux is sufficiently large. We show that in order to ignite a self-sustained combustion wave, sufficient heat must be supplied, so that in addition to providing a sufficiently high temperature to form a preheat layer, there is sufficient buoyant flux of oxidizer to allow the combustion wave to become self-sustained. In particular, combustion will not occur in microgravity environments unless there are other mechanisms of oxidizer transport. Some samples will not support a combustion wave because the flux required for ignition is larger than the critical level for extinction. Under nonadiabatic conditions, another extinction mechanism exists in which heat loss lowers the temperature below a critical level. The results of our analysis compare favorably with experimental observations of downward buoyant combustion in the region away from the ends of the sample.