Flows in complex geometries, such as porous media or biological networks, often contain plugs of liquid flowing within air bubbles. These flows can be modeled in microfluidic devices in which the geometric complexity is well defined and controlled. We study the flow of wetting liquid plugs in a bifurcating network of micro-channels. In particular, we focus on the process by which the plugs divide as they pass each bifurcation. The key events are identified, corresponding to large modifications of the interface curvature, the formation of new interfaces, or the division of a single interface into two new ones. The timing of the different events and the amplitude of the curvature variations are analyzed in view of the design of an event-driven model of flow in branching micro-networks. They are found to collapse onto a master curve dictated by the network geometry.