We present spatially resolved kinematics of 31 ALMA-identified dust-obscured star-forming galaxies (DSFGs) at 𝑧 ∼ 1.3-2.6, as traced by H𝛼 emission using VLT/KMOS near-infrared integral field spectroscopy from our on-going Large Programme "KMOS-ALMA Observations of Submillimetre Sources" (KAOSS). We derive H𝛼 rotation curves and velocity dispersion profiles for the DSFGs. Of the 31 sources with bright, spatially extended H𝛼 emission, 25 display rotation curves that are well fit by a Freeman disc model, enabling us to measure a median inclination-corrected velocity at 2.2 𝑅 d of 𝑣 rot = 190 ± 30 km s −1 and a median intrinsic velocity dispersion of 𝜎 0 = 87 ± 6 km s −1 for these disc-like sources. By comparison with less actively starforming galaxies, KAOSS DSFGs are both faster rotating and more turbulent, but have similar 𝑣 rot /𝜎 0 ratios, median 2.4 ± 0.5. We suggest that 𝑣 rot /𝜎 0 alone is insufficient to describe the kinematics of DSFGs, which are not kinematically "cold" discs, and that the individual components 𝑣 rot and 𝜎 0 indicate that they are in fact turbulent, but rotationally supported systems in ∼ 50 per cent of cases. This turbulence may be driven by star formation or mergers/interactions. We estimate the normalisation of the stellar Tully-Fisher relation (sTFR) for the disc-like DSFGs and compare it with local studies, finding no evolution at fixed slope between 𝑧 ∼ 2 and 𝑧 ∼ 0. Finally, we use kinematic estimates of DSFG halo masses to investigate the stellar-to-halo mass relation, finding our sources to be consistent with shock heating and strong feedback which likely drives the declining stellar content in the most massive halos.