KAOSS: turbulent, but disc-like kinematics in dust-obscured star-forming galaxies at $z\sim$1.3-2.6

We present spatially resolved kinematics of 27 ALMA-identified dust-obscured star-forming galaxies (DSFGs) at $z\sim$1.3-2.6, as traced by H$\alpha$ emission using VLT/KMOS near-infrared integral field spectroscopy from the "KMOS-ALMA Observations of Submillimetre Sources" (KAOSS) Large Programme. We derive H$\alpha$ rotation curves and velocity dispersion profiles for the DSFGs, and find that among the 27 sources with bright, spatially extended H$\alpha$ emission, 24 display evidence for disc-like kinematics. We measure a median inclination-corrected velocity at 2.2$R_{\rm d}$ of $v_{\rm rot}=$190$\pm$40kms$^{-1}$ and intrinsic velocity dispersion of $\sigma_0=$87$\pm$6kms$^{-1}$ for these disc-like sources. The kinematics yield median circular velocities of $v_{\rm circ}=$230$\pm$20kms$^{-1}$ and dynamical masses within 2$R_{\rm e}$ ($\sim$7kpc radius) of $M_{\rm dyn}=$(1.1$\pm$0.2)$\times$10$^{11}$M$_\odot$. Compared to less actively star-forming galaxies, KAOSS DSFGs are both faster rotating with higher intrinsic velocity dispersions, but have similar $v_{\rm rot}/\sigma_0$ ratios, median $v/\sigma_0=$2.5$\pm$0.5. We suggest that the kinematics of the DSFGs are primarily rotation supported but with a non-negligible contribution from pressure support, which may be driven by star formation or mergers/interactions. We estimate the normalisation of the stellar mass Tully-Fisher relation (sTFR) for the disc-like DSFGs and compare it with local studies, finding no evolution at fixed slope between $z\sim$2 and $z\sim$0. Finally, we show that the kinematic properties of the DSFG population are consistent with them evolving into massive early-type galaxies, the dominant $z\sim$0 population at these masses.