This paper presents a general explicit time integration scheme for dynamics simulations using the eXtended Finite Element Method with standard critical time step. We use the generalized mass lumping technique proposed in part I of this paper. This technique allows us to consider arbitrary enrichment functions in the X-FEM for explicit dynamics simulations. In this second part, the proposed approach allows the use of standard finite elements critical time step estimates. For that purpose, we developp a classical element-by-element strategy that couples the standard central difference scheme with the unconditionally stable explicit scheme proposed by Chang ({\em International Journal for Numerical Methods in Engineering}, 77(8):1100-1120, 2008). This scheme coupled with X-FEM allows us to recover FE critical time steps independently of the enrichment functions considered. Furthermore, a study of the stability property of this new explicit scheme is proposed in the X-FEM framework, both with fixed enrichments and evolving enrichments with time. Some examples illustrate the good properties of the new explicit numerical time scheme and some applications to dynamics crack growth are given.