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Gravitational waves may be one of the few direct observables produced by ultralight bosons, conjectured dark matter candidates that could be the key to several problems in particle theory, high-energy physics and cosmology. These axionlike particles could spontaneously form "clouds" around astrophysical black holes, leading to potent emission of continuous gravitational waves that could be detected by instruments on the ground and in space. Although this scenario has been thoroughly studied, it has not been yet appreciated that both types of detector may be used in tandem (a practice known as "multibanding"). In this paper, we show that future gravitational-wave detectors on the ground and in space will be able to work together to detect ultralight bosons with masses $25 \lesssim μ/\left(10^{-15}\, \mathrm{eV}\right)\lesssim 500$. In detecting binary-black-hole inspirals, the LISA space mission will provide crucial information enabling future ground-based detectors, like Cosmic Explorer or Einstein Telescope, to search for signals from boson clouds around the individual black holes in the observed binaries. We lay out the detection strategy and, focusing on scalar bosons, chart the suitable parameter space. We study the impact of ignorance about the system's history, including cloud age and black hole spin. We also consider the tidal resonances that may destroy the boson cloud before its gravitational signal becomes detectable by a ground-based follow-up. Finally, we show how to take all of these factors into account, together with uncertainties in the LISA measurement, to obtain boson mass constraints from the ground-based observation facilitated by LISA.