Over the past years, the equilibrium sorption of gases in polymers has been intensively studied. Mostly, glassy polymers were investigated because of their excellent selective mass transport properties. This work does not focus on the equilibrium sorption but on the kinetics to reach the equilibrium. We developed a new experimental method measuring the sorption-induced dilation kinetics of a polymer film. Carbon dioxide and glassy, aromatic polyimides were chosen as model systems. Low-pressure experiments demonstrate that the measured dilation kinetics represent the sorption kinetics. A significant delay between the sorption and dilation kinetics is based on the fact that dilation kinetics occurs simultaneously with the concentration increase in the center of the polymer film. High-pressure experiments reveal significant differences in dilation kinetics compared to low-pressure experiments. Generally, three regimes can be distinguished in the dilation kinetics : a first, fast volume increase followed by two much slower regimes of volume increase. The magnitude of fast and slow dilation kinetics strongly depends on the swelling history of the polymer sample. The results of the experiments are analyzed in the light of a model relating the fast dilation kinetics to a reversible "Fickian" dilation and the slower dilation kinetics to an irreversible, relaxational dilation.