In this paper, we present a discontinuous Galerkin finite element method for the solution of the transient Stokes equations on moving domains. For the discretization, we use an interior penalty Galerkin approach in space, and an upwind technique in time. The method is based on a decomposition of the space-time cylinder into finite elements. Our focus lies on three-dimensional moving geometries, thus we need to triangulate four dimensional objects. For this, we will present an algorithm to generate (d + 1)-dimensional simplex space-time meshes, and we show under natural assumptions that the resulting space-time meshes are admissible. Further, we will show how one can generate a four-dimensional object resolving the domain movement. First numerical results for the transient Stokes equations on triangulations generated with the newly developed meshing algorithm are presented.