Nonlinear stability is an essential characteristic that enhances the robustness and reliability of modern high-order numerical methods. The compressible Navier–Stokes (CNS) equations are equipped with a convex mathematical entropy function that is related to thermodynamic entropy. This function acts as an admissibility criterion for identifying physically relevant weak solutions and provides a means of demonstrating nonlinear (entropy) stability by bounding in the L2 norm the solution. Over the past several decades, the entropy stability of the CNS equations has received extensive attention and study. In practical engineering applications, the ability to dynamically move or adapt a mesh is fundamental, particularly in scenarios involving fluid-structure interaction or dynamic problems. Recent advancements in the field have established the entropy stability under grid deformation by adopting a discontinuous Galerkin split form of the arbitrary Lagrangian Eulerian (ALE) formulation. This work extends and enhances earlier findings concerning the implementation of solid wall boundary conditions for the CNS equations. A significant contribution of this study is the proof of entropy conservation for arbitrary wall motion, which is essential for simulating scenarios where boundary conditions can change in a deterministic manner or in response to fluid flow or external forces. The theoretical results demonstrate entropy conservation at both the continuous and semi-discrete levels, achieving accuracy up to machine precision in computational implementation. Moreover, additional dissipation, achieved through a specific upwinding technique, leads to entropy-stable results that satisfy both the requirements of physical admissibility and computational stability. Based on these theoretical results, several test cases are discussed, ranging from verification tests that evaluate the accuracy and convergence of the proposed numerical methods to established validation tests that benchmark performance against known solutions or experimental data. Finally, real-world applications are presented, showcasing the versatility and robustness of the developed methods.