Water sorption from the air is a ubiquitous mass transfer phenomenon with important applications in thermal energy storage, dehumidification, passive cooling, and atmospheric water harvesting. Numerous hygroscopic materials have been developed targeting high water vapor uptake and fast sorption kinetics. In particular, hygroscopic hydrogels have recently emerged as low-cost, high-performance sorption materials with the potential for high equilibrium vapor uptake and fast kinetics. However, the mechanisms behind hydrogel equilibrium and kinetics remain highly unexplored, which hinders their rational design and optimization for practical use. In this work, we study the impact of the nanoscale polymer network in the macroscopic sorption of hydrogels. We synthesize polyacrylamide-lithium chloride hydrogel composites with different crosslinking properties and, therefore, different polymer networks. When swelling these hydrogels in liquid, aqueous lithium chloride solutions, we find that both the equilibrium swelling ratio and swelling kinetics are significantly affected by changes in the crosslinking density. Surprisingly, we observe that crosslinking properties do not affect sorption, i.e., the uptake of water vapor. Instead, we find that at controlled humidity and temperature sorption equilibrium and sorption kinetics are independent of crosslinking properties. We elucidate these results by considering the thermodynamic properties of the lithium chloride-water vapor equilibria and the nonlinear diffusion of liquids inside hydrogels, showing that the hygroscopic properties of lithium chloride determine both equilibrium and kinetics. This work extends the fundamental understanding of the mass transport mechanisms in hygroscopic hydrogels and provides critical guidance for their practical design which can enable their use as next-generation sorption-based thermal energy storage systems, air dehumidifiers, passive thermal management devices, and atmospheric water harvesters.