Within the endeavour of understanding and tackling problems like delays and their propagation, or the optimisation of operations, air transport has customarily been studied through microscale dynamical models, describing the movements of its constituting elements according to sets of pre-hoc hypotheses and rules. The relevance of such models is nevertheless bounded by the realism and completeness of such rules. We here propose a complementary information-theoretic approach that does not rely on any pre-assumed model, but instead treats airports as information processing units. This allows to investigate the dynamics of airports in terms of information processing, whereby the relationship between the different aspects of their operations is expressed in terms of information contained, shared and transferred. Leveraging techniques from information decomposition, we describe such relationships in a large data set covering operations in Europe and US, focusing on departures and arrivals at the largest airports therein. Contrary to standard expectations, we find that departure dynamics is not a direct function of arrival one; departure delay shows a prominent dependency on the saturation of airports; and that there is a complex relationship between the way airports process information and their size, across both US and EU, with some notable exceptions. We further discuss the challenges appearing when this approach is used to assess the temporal evolution of the system, the information synergies and redundancies between different aspects of operations, and its integration with other standard models towards the evaluation of new policies and procedures.