The material point method (MPM) is gaining increasing amounts of attention due to its capacity to solve geotechnical problems involving large-deformations. While some problems require dynamic analysis, simulating the (infinite) continuous domain using typical Dirichlet (fixed) boundary conditions induce spurious reflections causing (1) unrealistic stress increments at the domain oundary and (2) the appearance of multiple unnatural stress waves in the domain. Aiming to eliminate this numerical artifact in MPM, two solutions for absorbing boundary conditions found in FEM are implemented and investigated; these are (1) a viscous boundary condition and (2) a viscoelastic boundary condition. The use of such dynamic boundary conditions in MPM is scarce and no validation of them has yet been presented in the literature. In this work, these absorbing conditions are implemented alongside other recent developments, which improves the numerical stability (Double-Mapping, Generalized Material Point Method, Composite Material Point Method), using two approaches: (1) directly imposing at the external active boundary nodes and (2) imposing via shape function interpolation. The proposed solutions are then validated with a one-dimensional benchmark: a soil column under dynamic load in small-deformation and large-deformation, and a 2D symmetric plane strain model under one loading pulse. The benchmark results demonstrate that the numerical reflections that lead to inaccuracies of stress and velocity can be removed using GIMP interpolation and, together with the other numerical technics, render high-quality and realistic results. A study of shallow foundation failure under repeated loading is also presented, showing the potential of applications of the proposed solution for modelling extreme geotechnical events.