The linear dynamics of perturbations developing in a channel with compliant walls is investigated for pulsatile base flows. It is found that the flow dynamics is mainly governed by four control parameters: the Reynolds number Re, the reduced velocity , the Womersley number Wo and the amplitude of the base flow modulation. Linear stability analyses are carried out within the framework of Floquet theory, implementing an efficient approach for removing spurious eigenmodes. The characteristics of flow-based (Tollmien-Schlichting) and wall-based (both travelling-wave flutter and divergence) modes are investigated over a large control-parameter space. It is shown that travelling-wave flutter (TWF) modes are predominantly influenced by the reduced velocity and that the Reynolds number has only a marginal effect. The critical reduced velocity (corresponding to onset of linear instability) is demonstrated to depend both on the Womersley number and modulation amplitude for a given set of wall parameters. The Tollmien-Schlichting (TS) mode is only weakly affected by the flexibility of the wall. Finally, the classification given by Benjamin (J. Fluid Mech. 16 436-450, 1963) is found to be too restrictive in the case of pulsatile base flows. In particular, a new type of transition mode is identified that results from the coalescence of two Floquet eigenmodes: interaction between TS and TWF modes due to coupling of the different Floquet harmonics, a phenomenon specific to time-periodic base flows.