Motivated to understand the process of melt migration in the earth’s mantle, we have studied a generalised form of Darcy’s law that describes porous flow in a matrix that can deform by creep. We find a remarkable richness of phenomena, including a new class of solitons. These consist of shape‐preserving waves of high liquid fraction which buoyantly ascend through a stationary matrix. This may have important implications for the morphology and geochemistry of primary igneous processes, and applicability to other porous flow problems.