A general analysis for grain-boundary electromigration applicable for type-A, -B, and -C diffusion kinetics is presented. Following Whipple's approach in analyzing grain-boundary diffusion, solutions for composition profiles have been derived for isolated boundaries (for type-B and -C kinetics) and for interacting boundaries (for type-A kinetics). Asymptotic expressions for the exact solution have been obtained using the method of steepest descents and conditions for their validity are specified. The asymptotic solutions with opposite driving forces can be linearly combined into two relationships which relate the slopes of the composition profiles directly to the grain-boundary diffusivity and the effective charge. These relationships provide useful formulas to analyze the edge profiles measured in cross-stripe experiments. Martin's solution was found to provide a reasonable approximation for the composition profile within the valid range of the asymptotic solutions.