We develop a theory for the dynamics of an interface in a two-phase elastic solid with kinetics driven by mass transport and stress. We consider a two-phase system consisting of bulk regions separated by a sharp interface endowed with energy and capable of supporting force. Our discussion is based on balance laws for mass and force in conjunction with a version of the second law-appropriate to a mechanical system out of equilibrium-which we use to develop a suitable constitutive theory for the interface. It is assumed that mass transport is characterized by the bulk diffusion of a single independent species; we neglect mass diffusion within the interface; limit our discussion to a continuous chemical potential and to a coherent interface; neglect the elasticity of the interface; and consider only infinitesimal deformations, neglecting inertia. We show that the field equations and free-boundary conditions can be developed in a simple manner in terms of the diffusion potential and its time derivatives, as opposed to the usual formulation in terms of concentration. Natural consequences of the thermodynamic framework are Lyapunov functions for the resulting evolution problems. This leads to a hierarchy of variational principles that should describe the equilibrium shapes of misfitting particles as well as possible microstructures that might form; these principles are applicable both in the absence and presence of an applied stress.