We develop a general thermodynamical description of an evolving interf
ace appropriate to situations far from equilibrium. The theory represe
nts a broad departure from theories based on classical nonequilibrium
thermodynamics, as we do not assume a linear relationship between flux
es 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 includ
ed as a special case; and we allow for heat flow in both phases. As an
application of the general theory, we develop linearized interface co
nditions for an evolving interface.