We consider a hybrid compressible/incompressible system with memory effects introduced by Lefebvre Lepot and Maury (2011) for the description of one-dimensional granular flows. We prove a first global existence result for this system without additional viscous dissipation. Our approach extends the one by Cavalletti, Sedjro, Westdickenberg (2015) for the pressureless Euler system to the constraint granular case with memory effects. We construct Lagrangian solutions based on an explicit formula of the monotone rearrangement associated to the density and explain how the memory effects are linked to the external constraints imposed on the flow. This result is finally extended to a heterogeneous maximal density constraint depending on time and space.