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The emergent collective dynamics in phase-separated mixtures of isometric active and passive Brownian particles is studied numerically in two-dimensions. A novel steady-state of well-defined propagating interfaces is observed, where the interface between the dense and the dilute phase propagates and the bulk of both phases is (nearly) at rest. Two kind of interfaces, advancing and receding, are formed by spontaneous symmetry breaking, induced by an instability of a planar interface due to the formation of localized vortices. The propagation arises due to flux imbalance at the interface, resembling the growth behavior of rough surfaces far from equilibrium. Above a threshold, the interface velocity decreases linearly with increasing fraction of active particles.