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We present a generalised architecture for reactive mobile manipulation while a robot's base is in motion toward the next objective in a high-level task. By performing tasks on-the-move, overall cycle time is reduced compared to methods where the base pauses during manipulation. Reactive control of the manipulator enables grasping objects with unpredictable motion while improving robustness against perception errors, environmental disturbances, and inaccurate robot control compared to open-loop, trajectory-based planning approaches. We present an example implementation of the architecture and investigate the performance on a series of pick and place tasks with both static and dynamic objects and compare the performance to baseline methods. Our method demonstrated a real-world success rate of over 99%, failing in only a single trial from 120 attempts with a physical robot system. The architecture is further demonstrated on other mobile manipulator platforms in simulation. Our approach reduces task time by up to 48%, while also improving reliability, gracefulness, and predictability compared to existing architectures for mobile manipulation. See https://benburgesslimerick.github.io/ManipulationOnTheMove for supplementary materials.