The initial electroosmotic flow through a small pore or microchannel with annular or rectangular cross section is studied under the Debye-Huckel approximation. Analytical series solutions and their asymptotic behavior for small and large non-dimensional electrokinetic widths are found for these two basic cases. The explicit and accurate solutions are particularly useful for examining various geometric/physical effects on the transient time scales and the flow rates for the transient states. The steady flow rate of the smaller channel may be disproportionately smaller than a reference channel if the electric double layer is thick, but will be in close proportion to the area ratio if the electric double layer is thin. A smaller channel compared to a reference channel has a shorter transient time scale, and the transient flow has characters very different from the steady state if the electric double layer is thin. The total transient flow rate of several smaller pores or channels may exceed largely that of a single large pore or channel with the same total cross section on the transient time scale of the smaller channels. The results have important implications on liquid transport in micropores or channels by pulse voltages or more general time-varying voltages.