A novel strategy to solve the finite volume discretization of the unsteady Euler equations within the Arbitrary Lagrangian–Eulerian framework over tetrahedral adaptive grids is proposed. The volume changes due to local mesh adaptation are treated as continuous deformations of the finite volumes and they are taken into account by adding fictitious numerical fluxes to the governing equation. This peculiar interpretation enables to avoid any explicit interpolation of the solution between different grids and to compute grid velocities so that the Geometric Conservation Law is automatically fulfilled also for connectivity changes. The solution on the new grid is obtained through standard ALE techniques, thus preserving the underlying scheme properties, such as conservativeness, stability and monotonicity. The adaptation procedure includes node insertion, node deletion, edge swapping and points relocation and it is exploited both to enhance grid quality after the boundary movement and to modify the grid spacing to increase solution accuracy. The presented approach is assessed by three-dimensional simulations of steady and unsteady flow fields. The capability of dealing with large boundary displacements is demonstrated by computing the flow around the translating infinite- and finite-span NACA 0012 wing moving through the domain at the flight speed. The proposed adaptive scheme is applied also to the simulation of a pitching infinite-span wing, where the bi-dimensional character of the flow is well reproduced despite the three-dimensional unstructured grid. Finally, the scheme is exploited in a piston-induced shock-tube problem to take into account simultaneously the large deformation of the domain and the shock wave. In all tests, mesh adaptation plays a crucial role.