The MZ relation between stellar mass (M-*) and metallicity (Z) of nearby galaxies has been described as both a global and local property, i.e. valid also on sub-galaxy scales, Here, we show that Z has remained a local property, following the gravitational potential, since z = 3. In absorption, the MZ relation has been well studied, and was in place already at z = 5.1. A recent absorption study of gamma-ray burst galaxies revealed a close match to damped Ly alpha (DLA) galaxies, surprising due to their vastly different impact parameters and leading the authors to suggest that local metallicity follows the local gravitational potential. In this paper, we formulate an observational test of this hypothesis. The test, in essence, forms a prediction that the velocity dispersion of the absorbing gas in galaxy haloes,normalized by the central velocity dispersion, must follow a steep log scale slope of -0.015 dex kpc(-1) as a function of impact parameter out to at least 20-30 kpc. We then compile an archival data and literature-based sample of galaxies seen in both emission and absorption suitable for the test, and find that current data confirm the hypothesis out to 40-60 kpc. In addition, we show that the distribution of the velocity offsets between z(em) and z(abs) favours a model where DLA systems are composed of individual sub-clouds distributed along the entire path through the halo, and disfavours a model where they are one single cloud with a bulk motion and internal sub-structure.