As robots become more versatile, they are increasingly found to operate together in the same environment where they must coordinate their motion in a distributed manner. Such operation does not present problems if the motion is quasi-static and collisions can be easily avoided. However, when the robots follow second-order dynamics, the problem becomes challenging even for a known environment. The setup in this work considers that each robot replans its own trajectory for the next replanning cycle. The planning process must guarantee the robot’s safety by ensuring collision-free paths for the considered period and by not bringing the robot to states where collisions cannot be avoided in the future. This problem can be addressed through communication among the robots, but it becomes complicated when the replanning cycles of the different robots are not synchronized and the robots make planning decisions at different time instants. This paper shows how to guarantee the safe operation of multiple communicating second-order vehicles, whose replanning cycles do not coincide, through an asynchronous, distributed motion planning framework. The method is evaluated through simulations, where each robot is simulated on a different processor and communicates with its neighbors through message passing. The simulations confirm that the approach provides safety in scenarios with up to 48 robots with second-order dynamics in environments with obstacles, where collisions occur often without a safety framework.