The objective of the present work is to develop a new numerical framework for simulations including moving bodies, in the specific context of high-order meshes consistent with Computer-Aided Design (CAD) representations. Thus, the proposed approach combines ideas from isogeometric analysis, able to handle exactly CAD-based geometries, and Discontinuous Galerkin (DG) methods with an Arbitrary Lagrangian-Eulerian (ALE) formulation , able to solve complex problems with moving grids. The resulting approach is a DG method based on rational Bézier elements, that can be easily constructed from Non-Uniform Rational B-Splines (NURBS), formulated in a general ALE setting. We focus here on applications in compressible aerodynamics, but the method could be applied to other models. Two verification exercises are conducted, to assess rigorously the properties of the method and the convergence rates for representations up to sixth order. Finally, two problems are analysed in depth, involving compressible Euler and Navier-Stokes equations , for an oscillating cylinder and a pitching airfoil. In particular, the convergence of flow characteristics is investigated, as well as the impact of using curved boundaries in the context of moving bodies.