Accurate and efficient aeroelastic models are critically important for enabling the optimization and control of highly flexible aerospace structures, which are expected to become pervasive in future transportation and energy systems. Advanced materials and morphing wing technologies are resulting in next-generation aeroelastic systems that are characterized by highly coupled and nonlinear interactions between the aerodynamic and structural dynamics. In this work, we leverage emerging data-driven modelling techniques to develop highly accurate and tractable reduced-order aeroelastic models that are valid over a wide range of operating conditions and are suitable for control. In particular, we develop two extensions to the recent dynamic mode decomposition with control (DMDc) algorithm to make it suitable for flexible aeroelastic systems: (1) we introduce a formulation to handle algebraic equations, and (2) we develop an interpolation scheme to smoothly connect several linear DMDc models developed in different operating regimes. Thus, the innovation lies in accurately modelling the nonlinearities of the coupled aerostructural dynamics over multiple operating regimes, not restricting the validity of the model to a narrow region around a linearization point. We demonstrate this approach on a high-fidelity, three-dimensional numerical model of an airborne wind energy system, although the methods are generally applicable to any highly coupled aeroelastic system or dynamical system operating over multiple operating regimes. Our proposed modelling framework results in real-time prediction of nonlinear unsteady aeroelastic responses of flexible aerospace structures, and we demonstrate the enhanced model performance for model predictive control. Thus, the proposed architecture may help enable the widespread adoption of next-generation morphing wing technologies.