Transport mechanisms involved in capillary condensation of water menisci in nanoscopic gaps between hydrophilic surfaces are investigated theoretically and experimentally by atomic force microscopy (AFM) measurements of capillary force. The measurements showed an instantaneous formation of a water meniscus by coalescence of the water layers adsorbed on the AFM tip and sample surfaces, followed by a time evolution of meniscus toward a stationary state corresponding to thermodynamic equilibrium. This dynamics of the water meniscus is indicated by time evolution of the meniscus force, which increases with the contact time toward its equilibrium value. Two water transport mechanisms competing in this meniscus dynamics are considered: (1) Knudsen diffusion and condensation of water molecules in the nanoscopic gap and (2) adsorption of water molecules on the surface region around the contact and flow of the surface water toward the meniscus. For the case of very hydrophilic surfaces, the dominant role of surface water transportation on the meniscus dynamics is supported by the results of the AFM measurements of capillary force of water menisci formed at sliding tip-sample contacts. These measurements revealed that fast movement of the contact impedes on the formation of menisci at thermodynamic equilibrium because the flow of the surface water is too slow to reach the moving meniscus.