We study speed of moving fronts in bistable spatially inhomogeneous media at parameter regimes where the speed tends to zero. We provide a set of conceptual assumptions under which we can prove power law asymptotics for the speed, with exponent depending a local dimension of the ergodic measure near extremal values. We also show that our conceptual assumptions are satisfied in a context of weak inhomogeneity of the medium and almost balanced kinetics, and compare asymptotics with numerical simulations. The presentation is based on a joint work with Arnd Sheel.