Moving groups in the solar neighborhood are ensembles of co-moving stars, likely originating from spiral-arm resonances, the Galactic bar, or external perturbations. Their co-movement with young clusters indicates recent star formation, but a lack of precise 3-D positions and velocities has obscured the connection. Using backward-orbit integrations of 509 clusters within ~1kpc -- based on Gaia DR3 and supplemented with APOGEE-2 and GALAH DR3 radial velocities -- we trace their evolution over the past 100Myr. Most clusters separate into three spatial branches that trace the Pleiades, Coma Berenices, and Sirius moving groups; no analogous branch is seen for the Hyades group. Clusters belonging to the Alpha Persei, Messier 6, and Collinder 135 families, previously shown to have formed in three massive star-forming complexes, commove with either the Pleiades (Alpha Persei, Messier 6) or Coma Berenices (Collinder 135). Our results sharpen the view of how large-scale Galactic dynamics shape nearby star formation. The cluster sample is drawn from the young-cluster catalogue of Hunt & Reffert (2023A&A...673A.114H, Cat. J/A+A/673/A114).