We develop a general thermodynamical description of an evolving interface appropriate to situations far from equilibrium. The theory represents a broad departure from theories based on classical nonequilibrium thermodynamics, as we do not assume a linear relationship between fluxes and forces, and we do not limit our theory to small departures from equilibrium. Further, we allow for diffusion in the bulk material; we allow the composition of material transferred across the interface to differ from the compositions of either phase, with solute drag included as a special case; and we allow for heat flow in both phases. As an application of the general theory, we develop linearized interface conditions for an evolving interface.