Robust robot motion planning in dynamic environments requires that actions be selected under real-time constraints. Existing heuristic search methods that can plan high-speed motions do not guarantee real-time performance in dynamic environments. Existing heuristic search methods for real-time planning in dynamic environments fail in the high-dimensional state space required to plan high-speed actions. In this paper, we present extensions to a leading planner for high-dimensional spaces, R*, that allow it to guarantee real-time performance, and extensions to a leading real-time planner, LSS-LRTA*, that allow it to succeed in dynamic motion planning. In an extensive empirical comparison, we show that the new methods are superior to the originals, providing new state-of-the-art search performance on this challenging problem.