Energy efficiency and bandwidth efficiency are two paramount important performance metrics for device-to-device communications. In this work, we investigate how mobility impacts EE and BE in a general framework of an LTEAdvanced network. First, we deploy a simple but practical mobility model to capture the track of the mobile devices. In particular, unlike previous works focusing on mobility velocity, which is difficult to obtain in practical mobile D2D systems, we deploy the parameter of device density to describe the device mobility. Next, we investigate the relationship between EE and BE in a mobile environment, and propose an EE-BE-aware scheduling scheme with a dynamic relay selection strategy that is flexible enough for making the transmission decision, including relay selection, rate allocation, and routing. Subsequently, through rigorous theoretical analysis, we derive a precise EE-BE trade-off curve for any device density and achieve the condition to attain the optimal EE and BE simultaneously. Finally, numerical simulation results are provided to validate the efficiency of the proposed scheduling scheme and the correctness of our analysis.