We used our database of VLT-UVES quasar spectra to build up a sample of 70 Damped Lyman-alpha (DLA) or strong sub-DLA systems with total neutral hydrogen column densities of log N(HI)>20 and redshifts in the range 1.72.43) and the lower (z_abs<2.43) redshift halves of our sample. However, the two populations of systems are statistically different. There is a strong redshift evolution in the sense that the mean metallicity and mean velocity width increase with decreasing redshift. We argue that the existence of a DLA velocity-metallicity correlation, over more than a factor of 100 spread in metallicity, is probably the consequence of an underlying mass-metallicity relation for the galaxies responsible for DLA absorption lines. Assuming a simple linear scaling of the galaxy luminosity with the mass of the dark-matter halo, we find that the slope of the DLA velocity-metallicity relation is consistent with that of the luminosity-metallicity relation derived for local galaxies. [...] abridged.