  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.
