Traditional covert communication often relies on the knowledge of the warden's channel state information, which is inherently challenging to obtain due to the non-cooperative nature and potential mobility of the warden. The integration of sensing and communication technology provides a promising solution by enabling the legitimate transmitter to sense and track the warden, thereby enhancing transmission covertness. In this paper, we develop a framework for sensing-then-beamforming in reconfigurable intelligent surface (RIS)-empowered integrated sensing and covert communication (ISACC) systems, where the transmitter (Alice) estimates and tracks the mobile aerial warden's channel using sensing echo signals while simultaneously sending covert information to multiple legitimate users (Bobs) with the assistance of RIS, under the surveillance of the warden (Willie). Considering channel estimation errors, we formulate a robust non-convex optimization problem that jointly designs the communication beamformers, the sensing signal covariance matrix at Alice, and the phase shifts at the RIS to maximize the covert sum rate of Bobs while satisfying the constraints related to covert communication, sensing, transmitter power, and the unit modulus of the RIS elements. To solve this complex problem, we develop an efficient algorithm using alternating optimization, successive convex approximation, S-procedure, sequential rank-one constraint relaxation, and semidefinite relaxation techniques. Numerical results confirm the convergence of the proposed algorithm and demonstrate its effectiveness in tracking the warden's channel while ensuring robust covert transmission. Furthermore, the results highlight the advantages of using RIS to enhance the covert transmission rate compared to baseline schemes, and also illustrate the intricate trade-off between communication and sensing in ISACC systems.