We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large scale dynamics of solutions to these equations is almost surely governed by a homogeneous Hamilton-Jacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general non-isotropic multidimensional setting. We also show that the rate of convergence of solutions to the Hamilton-Jacobi dynamics is at least algebraic in the relevant space-time scales when the initial data is close to an indicator function of a convex set. This talk is based on joint work with Andrej Zlato\v{s}.