MDAO frameworks like OpenMDAO and GEMSEO greatly facilitate setting up, solving and optimizing steady-state multiphysics problems. Unsteady problems, however, pose additional challenges since the time trajectory of the coupled problems needs to be traversed in reverse for adjoint gradient computations. RKOpenMDAO, an extension to OpenMDAO, enables the framework for time integration via diagonally implicit Runge-Kutta methods. It supports multidisciplinary reverse-mode sensitivity computations for MDAO in the unsteady domain. To implicitly solve the (nonlinear) Runge-Kutta stages of the multidisciplinary problems, quasi-steady nonlinear OpenMDAO problems are solved with block-preconditioned, Newton-Krylov-type solvers. In this approach, the component structure and parallelization capabilities of OpenMDAO are retained. The time integration is fully differentiated and available in both forward and reverse mode as long as all involved disciplinary components support the respective derivative modes. Thus, all the necessary capabilities for unsteady reverse-mode sensitivity analysis and optimization are provided. In this work, these time-accurate MDAO framework-capabilities are demonstrated and evaluated for selected testcases; among them a flow-control problem for a rotating cylinder attached to a tension spring in an unsteady viscous flow simulated with the CFD Software by ONERA, DLR and Airbus (CODA). In this example, the time-integrated aerodynamic drag functional is minimized w.r.t. the rotational speed of the cylinder. The correctness of the objective and its derivative are verified. Moreover, we intend to consider aerodynamic airfoil and wing-body aircraft configurations.