We consider unconfined spherical flames which expand into reactive gaseous premixtures that are laden with inert solid particles. An overall one-step Arrhenius reaction is adopted as burning process and the radiative transfer among the emitting/absorbing particles is modeled by a differential approximation of the Eddington type. The formulation accounts for the differences in velocity and temperature between the phases. The system is analyzed by matched asymptotic expansions in a multiple-limit process which assumes large Zel＇dovich numbers (activation to reaction temperature ratio); small Boltzmann numbers (radiant to convective heat flux ratio), small loading (solid to gas heat capacity ratio) by the particles and optically very thin flame fronts. Three main regions can then be distinguished in the flowfield: a thin reaction zone (dominated by molecular transports and reaction)embedded in a thicker quasi-steady flame-front region (dominated by molecular transports, convection and affected by conductive exchanges between the phases): the latter is itself flanked by even thicker, unsteady radiation-zones (dominated by radiative transfer and convection). We analytically show that, even for polydisperse particle-clouds, tabulating a one-parameter function once for all allows one to obtain an integral equation for the flame speed evolutions; the evolution equation is then solved numerically. Depending on the mixture and/or initial conditions, the interplays among nonlocal radiative-transfer, chemical nonlinearity, as well as momentum and conductive gas/particle exchanges, lead to smooth or abrupt flame accelerations, everlasting relaxation oscillations in fame speed or transient ones. Cases of trajectory multiplicity are also encountered.