In this talk, we will present a novel family of high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Discontinuous Galerkin (DG) schemes with a posteriori subcell Finite Volume (FV) limiter, on moving Voronoi tessellations that, at each time step, are regenerated in order to guarantee a high quality of the moving mesh. Remeshing may produce connectivity and topology changes: the main feature of our family of schemes is the capability of maintaining the high order of accuracy also in these situations. We achieve this result by a high order integration of the space-time conservation formulation of the governing hyperbolic PDE system not only over the extruded Voronoi elements, as in standard direct ALE schemes, but also over the sliver elements we build to fill the space–time holes originated where topology changes occur. This kind of schemes guarantee Galilean and rotational invariance and allow to better track contact discontinuities and moving material interfaces, thus they result very useful in multi-material applications and for the study of super/hyper-sonic objects in aerodynamics and astrophysics. In addition, astrophysical phenomena are characterized by the joint evolution of matter and spacetime which can be described by complex first order hyperbolic systems, like the Einstein field equations coupled with the general relativistic Euler equations. Here, in order to obtain stable and robust simulations, we have developed structure preserving schemes able to guarantee the exact preservation, up to machine precision, of equilibria and involution constraints. We will show some preliminary results for the simulation of neutron stars and accretion disks.