We present a reactive base control method that enables high performance mobile manipulation on-the-move in environments with static and dynamic obstacles. Performing manipulation tasks while the mobile base remains in motion can significantly decrease the time required to perform multi-step tasks, as well as improve the gracefulness of the robot's motion. Existing approaches to manipulation on-the-move either ignore the obstacle avoidance problem or rely on the execution of planned trajectories, which is not suitable in environments with dynamic objects and obstacles. The presented controller addresses both of these deficiencies and demonstrates robust performance of pick-and-place tasks in dynamic environments. The performance is evaluated on several simulated and real-world tasks. On a real-world task with static obstacles, we outperform an existing method by 48\% in terms of total task time. Further, we present real-world examples of our robot performing manipulation tasks on-the-move while avoiding a second autonomous robot in the workspace. See https://benburgesslimerick.github.io/MotM-BaseControl for supplementary materials.