We focus on the front propagation of diffusive flames obtained from the downward burning of inclined thermally thin solid fuels. This process consists of a pyrolysis reaction in the solid-phase and a combustion reaction in the gas phase. The solid-phase model is based on two coupled one-dimensional equations of temperature and solid density. We reduce the system into a single one-dimensional equation from which we obtain an analytical expression for the flame front speed. This expression may be understood as an upper bound of the burning spread rate in inclined samples. The gas-phase model is based on four coupled two-dimensional equations. These are employed to derive a criterion for determining the critical inclination angle beyond which the flame behavior becomes unstable. The comparison with the experiments confirms the validity of our predictions.