This paper investigates an aerial reconfigurable intelligent surface (RIS)-aided communication system under the probabilistic line-of-sight (LoS) channel, where an unmanned aerial vehicle (UAV) equipped with an RIS is deployed to assist two ground nodes in their information exchange. An optimization problem with the objective of maximizing the minimum average achievable rate is formulated to design the communication scheduling, the RIS's phase, and the UAV trajectory. To solve such a non-convex problem, we propose an efficient iterative algorithm to obtain its suboptimal solution. Simulation results show that our proposed design significantly outperforms the existing schemes and provides new insights into the elevation angle and distance trade-off for the UAV-borne RIS communication system.