This paper shows how a robot arm can follow and grasp moving objects tracked by a vision system, as is needed when a human hands over an object to the robot during collaborative working. While the object is being arbitrarily moved by the human co-worker, a set of likely grasps, generated by a learned grasp planner, are evaluated online to generate a feasible grasp with respect to both: the current configuration of the robot respecting the target grasp; and the constraints of finding a collision-free trajectory to reach that configuration. A task-based cost function enables relaxation of motion-planning constraints, enabling the robot to continue following the object by maintaining its end-effector near to a likely pre-grasp position throughout the object’s motion. We propose a method of dynamic switching between a local planner, where the hand smoothly tracks the object, maintaining a steady relative pre-grasp pose; and a global planner, which rapidly moves the hand to a new grasp on a completely different part of the object, if the previous graspable part becomes unreachable. Various experiments are conducted using a real collaborative robot and the obtained results are discussed.