We consider the CO oxidation in a continuous-flow reactor containing N catalytic layers of zeolite-supported palladium. For the kinetics of the reaction in one layer, we adopt a model proposed by Slin'ko and Jaeger. Rather than studying non-uniform coverage patterns on the individual catalyst layers, we focus on the influence of flow-mediated spatial coupling between the layers provided by variations in the CO partial pressure, which are transmitted by the flow to the adjacent downstream layer. Using the flow rate and the CO partial pressure at the inlet of the reactor as bifurcation parameters, in a parameter range where the reaction in one layer exhibits relaxational rate oscillations, we find different modes of operation for the reactor. The bifurcation diagram for two layers exhibits synchronized behavior at large flow rate. At lower flow rate and small CO partial pressure, we obtain a drop of catalytic activity at the first layer followed by a compensating increase at the second layer. In the mufti-layer system, an increasing number of layers works synchronously when the flow rate grows up. Then, downstream and upstream moving pulse trains of catalytic activity can develop.