Thin films of "complex salts" (CS = ionic surfactants with polymeric counterions) have recently been shown to respond to humidity changes in ambient air by changing their liquid crystalline structure. We here report isotherms and kinetics of water sorption/desorption for similar to 10-100 mu m films of alkyltrimethylammonium polyacrylate CS, measured in a dynamic gravimetric vapor sorption instrument over a 0-95% relative humidity (RH) range. The sorption per ion pair was similar to that observed for common ionomers. A kinetic model for the water exchange is presented, assuming that the "external" transport between the vapor reservoir and the film surface is rate-determining. The model predicts that the water content, after a small stepwise change of the reservoir RH, should vary exponentially with time, with a time constant proportional to both the slope of the sorption isotherm and the film thickness. These predictions were confirmed for our films over large RH ranges, and the external mass transfer coefficient in our setup was calculated from the experimental data. Expressions derived for the Biot number (ratio of characteristic times for internal and external water transport) for the considered limiting case strongly indicate that external water transport should quite generally affect, or even dominate, the measured kinetics for similarly thin hydrated films.