We analyze the gas kinematics of damped Lya systems (DLAs) hosting high z gamma-ray bursts (GRBs) and those toward quasars (QSO-DLAs) focusing on threestatistics: (1) dv, the velocity interval encompassing 90% of the totaloptical depth, (2,3) Wsi and Wciv, the rest equivalent widths of the SiII1526 and CIV1548 transitions. The dv distributions of the GRB-DLAs and QSO-DLAs are similar, each has median dv~80km/s and a significant tail to several hundred km/s. This suggests comparable galaxy masses for the parent populations of GRB-DLAs and QSO-DLAs and we infer the average dark matter halo mass of GRB galaxies is <~10^{12} Msol. The unique configuration of GRB-DLA sightlines and the presence (and absence) of fine-structure absorption together give special insight into the nature of high z, protogalactic velocity fields. The data support a scenario where the dv statistic reflects dynamics in the interstellar medium (ISM) and Wsi traces motions outside the ISM (e.g. halo gas, galactic-scale winds). The Wsi statistic and gas metallicity [M/H] are tightly correlated, especially for the QSO-DLAs: [M/H]=a + b log(Wsi/1A) with a=-0.92+/-0.05 and b=-1.41+/-0.10. We argue that the Wsi statistic primarily tracks dynamical motions in the halos of high z galaxies and interpret this correlation as a mass-metallicity relation with very similar slope to the trend observed in local, low-metallicity galaxies. Finally, the GRB-DLAs exhibit systematically larger Wsi values (>0.5A) than the QSO-DLAs (<Wsi>~ 0.5A) which may suggest galactic-scale outflows contribute to the largest observed velocity fields.
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