The transient response of electrolyte solutions in a narrow capillary tube or slit to a step change in the applied electric field and/or pressure gradient is analytically studied. The electric double layer adjacent to the charged capillary wall may have an arbitrary thickness relative to the capillary radius. The electrostatic potential distribution on a cross section of the capillary is developed by solving the Poisson-Boltzmann equation, and the fluid velocity profile evolving with time after the external field is imposed is obtained from the analytical solution of a modified Navier-Stokes equation. Closed-form formulas for the transient flow rate, electro-osmotic velocity, electric current, and streaming potential in the capillary are also derived as functions of relevant parameters. The results demonstrate that the behavior of the transient electrokinetic flow in a capillary tube and in a capillary slit is similar; however, the rate of evolution of the flow in a tube with time is faster by a factor of about 2 than that in a slit with its half thickness equal to the tube radius.