Flows around moving bodies are characterized by highly unsteady physical phenomena, such as complex vortical wakes and, in the transonic and supersonic regimes, moving shocks. As a consequence, the computational grid has to be properly refined in order to accurately simulate all the different flow conditions. In this context, Adaptive Mesh Refinement (AMR) represents a promising approach to reduce computational costs allowing, at the same time, to have an evolving mesh that is capable of capturing and following all the physical features of the given problem.The goal of this work is to investigate a coupling technique that allows the use of Arbitrary Lagrangian- Eulerian (ALE) descriptions of motion combined with AMR. We discretize the equations of fluid mechanics with the Isogeometric Discontinuous Galerkin method, using the ALE formulation to take into account the mesh movement. Non-Uniform Rational B-Splines (NURBS) are employed to represent both the grid coordinates and velocities. Exploiting the hierarchical properties of NURBS, it is possible to obtain high-order smooth mesh deformations that are compatible with the quadtree-like mesh refinement procedure, allowing a seamless ALE-AMR coupling.The proposed approach is tested on the pitching NACA 0012 airfoil benchmark and the gain with respect to non-adaptive meshes is quantified. The impact of different error indicators on mesh refinement is evaluated as well.