When you reach with your straw for the final drops of a milkshake, the liquid forms a train of plugs that flow slowly initially because of the high viscosity. They then suddenly rupture and are replaced with a rapid airflow with the characteristic slurping sound. Trains of liquid plugs also are observed in complex geometries, such as porous media during petroleum extraction, in microfluidic two-phase flows, or in flows in the pulmonary airway tree under pathological conditions. The dynamics of rupture events in these geometries play the dominant role in the spatial distribution of the flow and in determining how much of the medium remains occluded. Here we show that the flow of a train of plugs in a straight channel is always unstable to breaking through a cascade of ruptures. Collective effects considerably modify the rupture dynamics of plug trains: Interactions among nearest neighbors take place through the wetting films and slow down the cascade, whereas global interactions, through the total resistance to flow of the train, accelerate the dynamics after each plug rupture. In a branching tree of microchannels, similar cascades occur along paths that connect the input to a particular output. This divides the initial tree into several independent subnetworks, which then evolve independently of one another. The spatiotemporal distribution of the cascades is random, owing to strong sensitivity to the plug divisions at the bifurcations. microfluidics | respiratory flow T he motion of liquid plugs through a connected network of channels may involve many degrees of freedom evolving via a similarly large number of interactions: each immiscible interface introduces a degree of freedom into the problem owing to its ability to deform and to move, whereas each connecting branch between different areas introduces an interaction path that allows the flow in one region to influence the behavior in other areas of the network. The resulting flow pattern determines, among other things, how water or oil is extracted from porous media (1-3), the imbibition of paper (4, 5), and the stability of flow in a microfluidic device (6, 7).Liquid-gas two-phase flows also occur in the pulmonary airway tree, which is constantly coated with a thin liquid film. When the thickness of this film increases beyond some limit, plugs of liquid may form (8, 9) and therefore occlude the flow of air to the distal branches. Evidence of such behavior has been observed in pathologies ranging from asthma (10, 11) to cystic fibrosis (12). Furthermore, liquid plugs may be used as means to deliver medical treatment into the lung, e.g., in surfactant replacement therapy (13), and these plugs were observed to go through complex divisions, breaking and reforming before reaching their intended target (14).The structure of the lung as a branching binary tree has motivated many studies on the motion of gas-liquid flows into bifurcating channels (15)(16)(17)(18)(19)(20), with numerical work also taking into account the elasticity of the pulmonary walls, (e.g refs. ...