A number of Continuum Mechanics Boundary Value Problems involve large displacements or even large transformations of history-dependent (e.g., elasto-plastic) materials whose Finite Element Method (FEM) simulations would require remeshing procedures that are both computationally expensive and intricate for what concerns the transport of hardening variables. Among the so-called particle methods that aim at proposing suitable alternatives to FEM in this aspect, the Material Point Method (MPM, [1]) stands as one prominent possibility. The MPM actually overcomes mesh distortion issues through its consideration of a double layer of spatial discretization that includes, first, a fixed mesh serving as a history-less computational grid for solving continuum mechanics equations and, second, a set of material points which carry material constitutive behaviour and move freely within the mesh. The MPM discretization procedure then enables one to compute on the mesh a discrete field of nodal acceleration which should eventually serve to displace material points. However, the back-and-forth mapping between mesh nodes and material points lacks the possibility for a unique definition and a great number of various motion integration schemes have been proposed in the literature, since the PIC and FLIP precursors. The contribution will recall the reasons for this intricate issue in any MPM simulation and illustrate the consequences of using one or another motion integration scheme for what concerns the (non-)conservative nature of the method, in connection with the lumped mass matrix approximation that is classically used in MPM implementations. Various examples will be proposed, from simple configurations with known reference results to the more realistic setting of the collapse of an initial column of a frictional material. Results will also demonstrate an unexpected direct influence of the numerical time step on the numerical results with any PIC-based integration scheme, at variance with other schemes [2]. [1] Sulsky D., Chen Z., and Schreyer H.L., A particle method for history-dependent materials, Computer Methods in Applied Mechanics and Engineering, 118(1-2):179–196, 1994. [2] Duverger S., Duriez J., Philippe P., and Bonelli S., Critical comparison of motion integration strategies and discretization choices in the Material Point Method, Archives of Computational Methods in Engineering, 32:1369–1397, 2025