In this paper, we consider an unmanned aerial vehicle (UAV) aided mobile relaying system under a buffer constraint at the relay node. We propose a new relaying protocol employing mixed free-space optical/radio frequency (FSO/RF) communication, i.e., the source-relay and relay-destination link utilize FSO communication and RF communication, respectively, under the buffer constraint which is required to consider practical relay system. We study the trajectory optimization problem of buffer-constrained UAV-relaying in order to maximize the end-to-end data throughput. Taking the conditions of the mixed FSO/RF systems (e.g., a full-duplex relaying network that works in decode-and-forward, atmospheric attenuation, transmit power, and bandwidth in FSO and RF links) into consideration, we characterize the channel and throughput models. Furthermore, corresponding to the buffer-aided relaying, we derive a limited buffer constraint regarding the state of the queue in the buffer of relay. We solve the optimal trajectory problem of the UAV to maximize the throughput of user terminal using quadratically constrained programming. As a result, we propose an iterative algorithm that efficiently finds a local optimum solution for the throughput maximization problems. Our numerical results show that proposed buffer-aided mobile relaying achieves 161.3% throughput gains compared to a static relaying scheme.