The numerical simulation of time-dependent compressible flow fields presents significant challenges due to the presence of dynamic flow structures and evolving geometries. Accurate and efficient resolution of such phenomena requires advanced spatial and temporal discretization strategies. These flows often involve large boundary deformations, moving solid bodies, or dynamic internal flow features that demand localized grid refinement or motion. Moreover, compressible flows frequently exhibit multiple time scales, e.g. in low-Mach, reacting or multi-phase flows, or even viscous and inter-phase drag forces. In body-fitted methods, mesh deformation or adaptation is crucial to accurately and efficiently resolve complex flow patterns. The Arbitrarily Lagrangian–Eulerian (ALE) formulation has proven to be an effective strategy to handle relative motion between the mesh and the flow, since, in principle, it allows a complete separation of the mesh velocity from the material velocity. However, different characteristic time scales make the time integration challenging. Fully explicit schemes are often limited by stringent stability constraints requiring an excessively small time step, while fully implicit schemes can be computationally expensive and might lead to global nonlinear systems to be solved. Semi-implicit or IMEX (Implicit-Explicit) time integration methods provide a promising alternative by treating fast and slow dynamics differently. This minisymposium focuses on body-fitted finite-volume and discontinuous Galerkin methods that specifically address these challenges. Topics include Lagrangian and ALE schemes, with their high-order extension, over moving, adaptive, or overset grids, and time integration strategies such as IMEX or semi-implicit formulations. Contributions exploring key numerical features such as conservation, stability, moving boundary conditions, computational efficiency, as well as practical implementation aspects, are welcome. Target applications may involve single- and multi-phase flows, supercritical and non-ideal fluids, multispecies and reacting flows.