We present a novel framework for motion planning in dynamic environments that accounts for the predicted trajectories of moving objects in the scene. We explore the use of composite signed-distance fields in motion planning and detail how they can be used to generate signed-distance fields (SDFs) in real-time to incorporate predicted obstacle motions. We benchmark our approach of using composite SDFs against performing exact SDF calculations on the workspace occupancy grid. Our proposed technique generates predictions substantially faster and typically exhibits an 81--97% reduction in time for subsequent predictions. We integrate our framework with GPMP2 to demonstrate a full implementation of our approach in real-time, enabling a 7-DoF Panda arm to smoothly avoid a moving robot.