This work aims at developing a high-order, fully conservative, discretization of sliding grids with applications to compressible flows at different regimes (from subsonic to supersonic). The proposed approach combines a discontinuous Galerkin formulation for Navier-Stokes equations with rational representations originating from Isogeometric Analysis, which allows to design a watertight and fully conservative sliding grid algorithm. A verification exercise is first carried out to rigorously establish the convergence rate of the method. Then, the accuracy and robustness are demonstrated for a flow around a pitching ellipse at different regimes. A comparison with a grid deformation technique and a sensitivity study with respect to the location of the sliding interface are also investigated. Finally, the flow around a vertical-axis wind turbine configuration is simulated to show the potentiality of the proposed approach to deal with more complex geometries.