  We present the $0.6<z<2.6$ evolution of the ionized gas velocity dispersion
in 175 star-forming disk galaxies based on data from the full KMOS$^{\rm 3D}$
integral field spectroscopic survey. In a forward-modelling Bayesian framework
including instrumental effects and beam-smearing, we fit simultaneously the
observed galaxy velocity and velocity dispersion along the kinematic major axis
to derive the intrinsic velocity dispersion $\sigma_0$. We find a reduction of
the average intrinsic velocity dispersion of disk galaxies as a function of
cosmic time, from $\sigma_0\sim45$ km s$^{-1}$ at $z\sim2.3$ to
$\sigma_0\sim30$ km s$^{-1}$ at $z\sim0.9$. There is substantial intrinsic
scatter ($\sigma_{\sigma_0, {\rm int}}\approx10$ km s$^{-1}$) around the
best-fit $\sigma_0-z$-relation beyond what can be accounted for from the
typical measurement uncertainties ($\delta\sigma_0\approx12$ km s$^{-1}$),
independent of other identifiable galaxy parameters. This potentially suggests
a dynamic mechanism such as minor mergers or variation in accretion being
responsible for the scatter. Putting our data into the broader literature
context, we find that ionized and atomic+molecular velocity dispersions evolve
similarly with redshift, with the ionized gas dispersion being $\sim10-15$ km
s$^{-1}$ higher on average. We investigate the physical driver of the on
average elevated velocity dispersions at higher redshift, and find that our
galaxies are at most marginally Toomre-stable, suggesting that their turbulent
velocities are powered by gravitational instabilities, while stellar feedback
as a driver alone is insufficient. This picture is supported through comparison
with a state-of-the-art analytical model of galaxy evolution.
