A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind-biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual-time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h-adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi-core performance of the proposed technique is also discussed through the examples provided.