We study flames expanding radially from the center of a spherical enclosure which is initially filled with a two-phase premixture: premixed reactive gases and an inert solid suspension. An overall Arrhenius rate is postulated for the burning process, and the radiant exchanges among the particles is assumed to follow an Eddington differential equation specialized to a grey continuous medium. Different temperatures and velocities are allowed for each phase. Transfers to the walls and heating by compression are also accounted for. The problem is analysed by asymptotic methods in a multiple-limit procedure which assumes large Zel＇dovich numbers (activation to reaction temperature ratio), small Boltzmann numbers (radiant to convective heat-flux ratio), small loadings by the particles, optically very thin flame fronts and specific-heat ratios in the gas phase that are close to unity. We analytically reduce the whole problem to a single integral equation for the burning speed as a function of the fireball radius. Upon numerical integration the evolution equation yields the front burning speed histories. It is shown that: a) Compression of the fresh medium often yields a dominant source of flame-speed increase, especially at the end of the propagation. It also tends to screen the radiant exchanges with the wall significantly. b) The optical properties of the wall only affect the propagation at the end of the process if the container is large. For small vessels, radiative preheating is thoroughly weak. c) Mainly through hydrodynamics and compression, the confinement modifies the flame trajectories qualitatively; e.g., it often favors the appearance of jumps in burning speed. In any case, quantitative modifications are brought about. An overall conclusion is that straightforward extrapolations from small-scale experiments in enclosures to unconfined explosions do not constitute a safe procedure.