We consider a hp-version space-time discontinuous Galerkin (DG) method for solving time-dependent problems on moving domains. The method employs a fitted mesh approach, where the mesh is adapted to conform to the evolving geometry of the domain at each time step. An agglomeration strategy is introduced for handling cut cells, improving numerical stability and performance. The approach combines high-order space-time discretization with hp-adaptivity to achieve optimal convergence rates for parabolic problems. The resulting method allows for precise mesh refinement in regions of interest, resolving steep solution gradients and reduced-regularity features. Numerical experiments indicate efficient performance and stable behaviour across representative test cases.