Aims. We investigate the nature of the classical low-velocity structures in the local velocity field, i.e. the Pleiades, Hyades, and Sirius moving groups. After using a wavelet transform to locate them in velocity space, we study their relation with the open clusters kinematically associated with them. Methods. By directly comparing the location of moving group stars in parallax space to the isochrones of the embedded clusters, we check whether, within the observational errors on the parallax, all moving group stars could originate from the ongoing evaporation of the associated cluster. Results. We conclude that, in each moving group, the fraction of stars making up the velocity-space overdensity superimposed on the background is higher than the fraction of stars compatible with the isochrone of the associated cluster. These observations thus favour a dynamical (resonant) origin for the Pleiades, Hyades, and Sirius moving groups.