The one-dimensional approximate equation in the rectangular Cartesian coordinates governing pressure-driven flow of a non-Newtonian fluid in a thin slit whose walls are porous, with the lower plate is stationary while the upper plate is moving with the constant velocity in the x-direction, is derived by accounting of the order of magnitudes of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The non-Newtonian fluid behavior is described by power-law fluid model and the one-dimensional governing equation for a power-law fluid flow under the influence of pressure gradient in the narrow gap between two closely spaced parallel porous plates, with the lower plate is stationary while the top plate is exposed to sudden acceleration with a constant velocity in the x-direction, is solved analytically for a Newtonian fluid case (n = 1) and numerically for various values of power-law index to determine the transient and thus the overall transient velocity distributions. The effects of the transverse mass suction/injection rates at the bottom porous plate on the pressure-driven flows of non-Newtonian fluids are examined for various values of time, dynamic pressure drops in the axial direction and power-law indexes. The results obtained from the present analysis are compared with the data available in the literature. (C) 2006 Elsevier B.V. All rights reserved.