In computational fluid dynamics (CFD) simulations of compressible flows around moving bodies, mesh adaptation is crucial to accurately track evolving flow features, such as moving shock waves, and follow boundary deformations. Given a mesh–solution pair at a given time step, traditional adaptation procedures identify grid elements requiring modifications, perform local connectivity changes (such as node insertion, deletion, and edge swapping), and interpolate the solution onto the new mesh to continue the simulation. However, solution interpolation can introduce numerical errors, which may accumulate over time, degrading both accuracy and conservativeness of the solution. This work describes a pathbreaking approach designed to eliminate explicit solution interpolation during unstructured mesh adaptation. We re-interpret local mesh changes as sequences of fictitious compressions and expansions of the control volumes composing the computational domain. These continuous deformations can be naturally embedded within the Arbitrary Lagrangian–Eulerian (ALE) framework [B. Re et al. J Comput Phys, 340, 2017]. Moreover, the Geometric Conservation Law (GCL) is satisfied by construction. We demonstrate the application of this strategy within a finite-volume context and its extension to residual distribution schemes, which can achieve arbitrarily high-order accuracy on unstructured grids [S. Colombo and B. Re. Comput Fluids, 239, 2022]. Regarding time integration, the absence of explicit interpolation facilitates the use of high-order BDF schemes, as the solution at previous time steps can be retrieved through the history of the degrees of freedom indices. To validate the proposed methodology, we present numerical results of Euler simulations around moving two- and three-dimensional bodies.