Accurate simulation of fluid flow on evolving domains poses substantial challenges and there is still a need for innovation in cost-efficient methods. We present techniques for the simulation of incompressible flows on evolving domains. We discuss the integration of DWR adaptivity to automatically refine meshes to increase the resolution of approximations towards goal quantities. To efficiently solve the arising large linear systems, we use multigrid methods. Therefore, our approach is based on three key methods: - Space-time finite element methods naturally capture multiphysics problems in a holistic framework. These features facilitate the use of goal oriented adaptivity. On time-dependent domains cut cells yield a facet of additional complexity. - Goal-oriented space-time adaptivity for the optimization of the space-time mesh, which then allows for the optimal allocation of computational resources. This approach is especially promising for time-dependent meshes as the background mesh carries computational overhead. DWR-based adaptivity helps to eliminate this. - Multigrid methods are well-known for their efficiency in solving large-scale problems. Their application in the context of evolving domains and DWR space-time adaptivity presents unique challenges and opportunities, which we address. The talk is dedicated to demonstrate how the combination of DWR adaptivity and multigrid methods can lead to improvements in computational efficiency. This synergy is especially potent in handling the complexities associated with evolving domains, where the dynamic nature of the problem can lead to increasing computational demands.