Transient diffusion in two-dimensional geometries is considered. It is shown how the spatial variation of the mass transfer-limited flux of a minor species varies with time from initially uniform to the nonuniform, steady-state distribution. Flux distributions on sinusoidal electrode and on a line electrode embedded in an otherwise insulating plane are considered. A boundary-element method is used to solve the problem in Laplace transform space, and the results are subsequently inverted into the time domain.