An EUV-selected sample of late-type stars has been observed for lithium abundances and radial velocity radiations. It is found that essentially all of the single stars have lithium abundances comparable to or greater than late-type stars in the Pleiades. These observations have confirmed the hypothesis that a high proportion of the most coronally active, late-type stars are Li-rich, because they are young, have not yet spun down, and consequently exhibit enhanced dynamo activity. From these data and other activity-selected samples in the literature, a Li-rich sample of single stars has been chosen, which, by comparison with open cluster Li abundances and rotation rates, is substantially younger than 300 Myr. The kinematics of this sample has been investigated, and from the radial velocities a one-dimensional dispersion of only 5.7 km s^-1 is found, about a centroid space motion extremely similar to that of a group of young open clusters and nearby B stars, known as the Local Association. Consideration of the space motions confirms that, if this low-mass Local Association can be represented by a three-dimensional velocity dispersion of about 10 km s^-1, then the majority of the Li-rich stars are members. The origin of the Local Association is still obscure, but a number of suggestions are put forward, including ejection of stars from nearby open clusters, the disruption of pre-existing clusters, or the dissolution of the unbound remnants from one or more star formation regions.