Mucus films coating the inner surface of the pulmonary airways in the human tracheobronchial tree perform the important task of capturing alien particles and pathogens from inhaled air. This configuration is associated with two hydrodynamical problems that are closely linked to different pulmonary diseases: (1) mucociliary clearance (MCC), i.e. the evacuation of mucus toward the trachea through the coordinated beating of cilia, which can break down due to a change in mucus rheology, e.g. in the case of cystic fibrosis, where the elastic response of mucus becomes dominant; (2) liquid plug formation due to the Plateau-Rayleigh instability, e.g. as a result of mucus hypersecretion in chronic obstructive pulmonary disease. In order to identify possible levers to avoid such catastrophic events, e.g. in the context of drug development, a reduced modelling of these flows, preferably at the scale of the entire tracheobronchial tree, can be a valuable asset. In this talk, I will present two modelling contributions for the above-introduced problems. In the first case, we developed a continuum model of MCC based on a Navier boundary condition due to Bottier et al. (PLoS Comp. Biol., 2017), which mimics momentum transfer from the beating cilia, and used this model to study the effect of mucus viscoelasticity on MCC. In the second case, we applied the weighted residual integral boundary layer (WRIBL) technique of Ruyer-Quil and Manneville (Eur. Phys. J. B, 2000), which relies on the long-wave approximation, to represent the formation of liquid plugs from annular liquid films. We then applied this model to predict plug formation across the conducting zone of the tracheobronchial tree, based on the lung architecture model of Weibel, and to assess the potential for epithelial cell damage due to the associated wall stresses.