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Trapping and storing light for arbitrary time lengths in open cavities is a major goal of nanophotonics, with potential applications ranging from energy harvesting to optical information processing. Unfortunately, however, the resonance lifetime of conventional open resonators remains finite even in the limit of vanishing material absorption, as a result of radiation loss. In this context, bound states in the continuum (BiCs) have provided a unique way to achieve unbounded resonance lifetimes despite the presence of compatible radiation channels. However, physical constraints such as reciprocity, linearity, and delay-bandwidth limits prevent the possibility to externally excite such ideal bound states and make them interact with broadband sources. Here, we overcome these limitations and theoretically demonstrate that subwavelength open resonators undergoing a suitable temporal modulation can efficiently capture a broadband incident wave into a nonradiating eigenmode of the structure, leading to the first example of a BiC that is accessible to broadband light. Our findings unveil the dynamic capabilities of bound states in the continuum and extend their reach and potential impact for different applications.