We present a space-time hierarchical solution concept for optimization problems governed by the time-dependent Stokes-- and Navier--Stokes system. Discretisation is carried out with finite elements in space and one-step-$/theta$-schemes in time. By combining a Newton solver for the treatment of the nonlinearity with a space-time multigrid solver for linear subproblems, we obtain a robust solver whose convergence behaviour is independent of the number of unknowns of the discrete problem and robust with regard to the considered flow configuration.