We analyze the gas kinematics of damped Lyα systems (DLAs) hosting high-z gamma-ray bursts (GRBs) and those toward quasars (QSO-DLAs), focusing on three statistics: (1) Δv$_90$, the velocity interval encompassing 90% of the integrated optical depth, and (2) W$_1526$ and (3) W$_1548$, the rest equivalent widths of the Si II 1526 and C IV 1548 transitions, respectively. The Δv$_90$ distributions of the GRB-DLAs and QSO-DLAs are similar; each has median Δv$_90$ ≈ 80 km s$^-1$ and a significant tail, extending to several hundred km s$^-1$. This suggests comparable galaxy masses for the parent populations of GRB-DLAs and QSO-DLAs, and we infer that the average dark matter halo mass of GRB galaxies is lesssim10$^12$ M$_☉$. The unique configuration of GRB-DLA sight lines 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 in which the Δv$_90$ statistic reflects dynamics in the interstellar medium (ISM) and W$_1526$ traces motions outside the ISM (e.g., halo gas and galactic-scale winds). The W$_1526$ statistic and gas metallicity [M/H] are tightly correlated, especially for the QSO-DLAs: [M/H] = a + blog(W$_1526$/1̊A) with a = -0.92 +/- 0.05 and b = 1.41 +/- 0.10. We argue that the W$_1526$ 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 W$_1526$ values (>0.5 Å) than the QSO-DLAs (langleW$_1528$rangle ≈ 0.5 r̊), which may suggest that galactic-scale outflows contribute to the largest observed velocity fields.