To simulate large, history-dependent material displacements, the Material Point Method (MPM) solves for the kinematics of Lagrangian material points being embedded with mechanical variables while moving freely within a fixed mesh. The MPM procedure makes use of the latter mesh as a computational grid, where the momentum balance equation with the acceleration field are first projected onto nodes, before material points can be moved. During that process, a number of different choices have been adopted in the literature for what concerns the computational definition of time increments of velocity and position, from the knowledge of nodal acceleration. An overview of these different motion integration strategies is herein proposed, with a particular emphasis on their impact onto the MPM conservative properties. Original results illustrate the discussion, considering either simple configurations of solid translation and rotation or a more complex collapse of a frictional mass. These analyses furthermore reveal hidden properties of some motion integration strategies regarding conservation, namely a direct influence of the time step value during a time integration being inspired by the Particle In Cell (PIC) ancestor of the MPM. The spatial, resp. temporal (in comparison with vorticity), discretizations are also shown to affect the angular momentum conservation of the FLIP method, resp. an affine extension of PIC (APIC).