We consider a $hp$-version space-time discontinuous Galerkin (DG) method for solving time-dependent problems on moving domains. Building on the space-time DG framework of Cangiani et al. (2017, SISC) and the theoretical results on essentially arbitrary polytopic elements in Cangiani et al. (2021, Math. Comp.), we extend the approach to evolving geometries by employing a fitted space-time mesh that conforms to the domain on each time slab. Cut cells generated by the moving boundary are handled via an agglomeration procedure, producing shape-regular polytopic elements in a boundary layer. The analysis accommodates curved element boundaries without resorting to nonlinear elemental mappings. The resulting high-order space-time discretisation supports $hp$-adaptivity and exhibits the expected convergence behaviour, while enabling local refinement. Numerical experiments demonstrate the method’s robustness and efficiency in scenarios involving large deformations. The proposed space-time DG method is shown to be flexible, efficient, and well-suited for a broad range of moving-domain problems.