In this paper, we investigate both experimentally and theoretically the dynamics of a liquid plug driven by a cyclic periodic forcing inside a cylindrical rigid capillary tube. First, it is shown that depending on the type of forcing (flow rate or pressure cycle), the dynamics of the liquid plug can either be stable and periodic, or conversely accelerative and eventually leading to the plug rupture. In the latter case, we identify the sources of the instability as: (i) the cyclic diminution of the plug viscous resistance to motion due to the decrease in the plug length and (ii) a cyclic reduction of the plug interfacial resistance due to a lubrication effect. Since the flow is quasi-static and the forcing periodic, this cyclic evolution of the resistances relies on the existence of flow memories stored in the length of the plug and the thickness of the trailing film. Second, we show that contrary to unidirectional pressure forcing, cyclic forcing enables breaking of large plugs in confined space though it requires longer times. All the experimentally observed tendencies are quantitatively recovered from an analytical model. This study not only reveals the underlying physics but also opens up the prospect for the simulation of "breathing" of liquid plugs in complex geometries and the determination of optimal cycles for obstructed airways reopening.