Usually time dependent evolution equations are solved time step by time step. In each step, a system of equations corresponding to the spatial discretization has to be solved. Such methods can be parallelized only in space, but the size of the spatial problem limits the strong scaling behavior. The use of multigrid methods, which treat multiple time steps in an all-at-once system, can significantly improve parallel scaling compared to a time-stepping application with geometric multigrid solvers [2, 3]. Due to the improved communication pattern between the parallel processes, this is also true when using a time-simultaneous multigrid method without temporal parallelization [1]. Here, we numerically analyze how multigrid waveform relaxation methods and space-time multigrid methods behave for convection-diffusion-reaction equations found in fluid flow problems as the problems become increasingly transport dominated. Some global-in-time solution strategies of the Oseen equations benefit from an Augmented Lagrangian (AL) acceleration but in each linear velocity subproblem a modified PDE has to be solved. Even in a time-sequential setting, special smoothers and transfer operators are required to efficiently handle the resulting ill-conditioned systems. We analyze how the convergence behavior of these space-time multigrid methods is affected by the AL approach. Furthermore, we consider problems with space- and time-dependent diffusion and convection parameters, that arise in global-in-time solution strategies of the Navier-Stokes equations with non-Newtonian fluids. [1] J. Dünnebacke, S. Turek, C. Lohmann, A. Sokolov, P. Zajac (2021). Increased space- parallelism via time-simultaneous newton-multigrid methods for nonstationary non- linear pde problems. Int. J. High Perform. Comput. Appl., 35(3), 211–225. [2] M.J. Gander, M. Neumüller (2014). Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems. SIAM J. Sci. Comput., 38(4), A2173 - A2208. [3] C. Lubich, A. Ostermann (1987). Multi-grid dynamic iteration for parabolic equations. BIT, 27(2), 216–234.