Diffusional mechanisms of electromigration and stress relaxation involve the flow of atoms in response to a gradient in chemical potential along an interface. This gradient in chemical potential may be provided by the component of an electric field parallel to the interface, or it may be established by the normal component of stresses along it. In either case, considerations of continuity of the potential dictate that diffusive flow must also be induced along any other boundary that intersects the interface. As an example, in this paper, a model system that contains grain boundaries normal to an applied electric field is analyzed. While the electric field does not directly induce diffusion along these grain boundaries, it is shown that a complimentary flux must be induced along them. The effect of this flux on electromigration is discussed in this paper. Furthermore, it is well-known that non-homogeneous diffusion of matter along boundaries induces elastic distortions and stress gradients. These in turn, influence the diffusion process. The effect of these elastic distortions on the atomic flux has been examined by considering diffusion along a single interface in an elastic medium. Prior studies of diffusional cavity growth have established the magnitudes of non-dimensional time-scales over which the deposition of atoms along the grain boundaries can be assumed to be essentially uniform. Such an assumption considerably simplifies analyses for stress evolution in these problems. The appropriate time-scales over which such a simplification can be made for electromigration are discussed in this paper, and illustrated by some model calculations.