As was found in our previous works, when Nafion swells in water, polymer fibers unwind into the bulk of the surrounding liquid. This effect is controlled by the content of deuterium in water. Here, we present the results of studying the dynamics of methylene blue (MB) adsorption on the Nafion surface for MB solutions based on natural water (deuterium content is 157 ppm, the unwinding effect occurs) and based on deuterium-depleted water (DDW; deuterium content is 3 ppm, there is no unwinding). In addition, we studied the dynamics of water desorption during drying of the Nafion polymer membrane after soaking in MB solution based on natural water and DDW. It turned out that in the case of natural water, the rate of MB adsorption and water desorption is higher than in the case of DDW. It also turned out that the amount of MB adsorbed on the membrane in the case of natural water is greater than in the case of DDW. Finally, it was found that the desorption of water during drying is accompanied by a rearrangement of the absorption spectrum of Nafion. This rearrangement occurs earlier in the case of DDW. Thus, by infinitesimal changes in the deuterium content (from 3 to 157 ppm) in an aqueous solution, in which a polymer membrane swells, we can control the dynamics of adsorption and desorption processes. A qualitative model, which connects the observed effects with the slowing down of diffusion processes inside the layer of unwound fibers, is proposed.