Active control of quantum systems enables diverse applications ranging from quantum computation to manipulation of molecular processes. Maximum speeds and related bounds have been identified from uncertainty principles and related inequalities, but such bounds utilize only coarse system information and loosen significantly in the presence of constraints and complex interaction dynamics. We show that an integral-equation-based formulation of conservation laws in quantum dynamics leads to a systematic framework for identifying fundamental limits to any quantum control scenario. We demonstrate the utility of our bounds in three scenarios-three-level driving, decoherence suppression, and maximum-fidelity gate implementations-and show that in each case our bounds are tight or nearly so. Global bounds complement local-optimization-based designs, illuminating performance levels that may be possible, as well as those that cannot be surpassed.