We address the surface tension-driven dynamics of porous media in nearly saturated pore-space solutions. We linearize this dynamics in the reaction-limited regime near its fixed points – surfaces of constant mean curvature (CMC surfaces). We prove that the only stable interface for this dynamics is the plane, and estimate the time scale for a CMC surface to become unstable. We also discuss the differences between open and closed system dynamics, pointing out the unlikelihood that CMC surfaces are ever realized in these systems on any time scale.