The traditional theory of transport across capillary membranes via a laminar Poiseuille flow is shown to be invalid. It is demonstrated that the random, diffusive nature of the molecular flow and interactions with the "pore" walls play an important role in the transport process. Neither the continuum Navier-Stokes theory nor the equivalent theory of irreversible thermodynamics is adequate to treat the problem. Combination of near-continuum hydrodynamic theory, noncontinuum kinetic theory, and the theory of fluctuations provides a first step toward modeling both liquid processes in general and membrane transport processes as a specific application.