Model order reduction (MOR) is essential for efficiently modeling and simulating high-dimensional complex systems. Manifold-based MOR has been extensively applied in dealing with steady-state dynamic responses of systems involving geometric nonlinearities. And it demonstrates excellent capability in reducing computational complexity while preserving accuracy. However, the application of this scheme to a flexible multi-body system (FMBS) remains limited due to the challenges in dealing with its transient dynamic property. Besides, an FMBS exhibits the strong coupling between overall rigid-body motion and large elastic deformations of flexible components, as well as the presence of kinematic constraints stemming from joints, sliders, etc. In this work, the MOR method based on quadratic manifold (QM) [1] is developed for efficient simulation of an FMBS. The coupling between overall rigid-body motion and local deformation is modeled using the floating frame of reference formulation. Subsequently, the Craig-Bampton method is employed to separate the constrained boundary degrees of freedom (DOFs) from the internal unconstrained ones, remaining constraint relations of the boundary DOFs while generating constraint modes for the internal DOFs. The QM, consisting of vibration modes and their corresponding modal derivatives [2], is then constructed to reduce the internal unconstrained coordinates involving geometric nonlinearity. Furthermore, the proposed method is extended to high-rotational-speed FMBSs by incorporating centrifugal stiffening and gyroscopic effects through a complex modal-based quadratic manifold. Accordingly, the reduced equations of motion are formulated and the solving procedure for QM-based MOR is presented. Finally, numerical benchmarks of FMBSs, including a rotating flexible beam, a flexible slider-crank mechanism, and a truss structure with kinematic constraints, as well as high-speed rotating configurations, demonstrates the higher efficiency and accuracy of the proposed method in comparison with other MOR methods of modal truncation and modal derivatives.