  We present the first statistical study of spatially integrated non-Gaussian
stellar kinematics spanning 7 Gyr in cosmic time. We use deep, rest-frame
optical spectroscopy of massive galaxies (stellar mass $M_\star > 10^{10.5}
{\rm M}_\odot$) at redshifts z = 0.05, 0.3 and 0.8 from the SAMI, MAGPI and
LEGA-C surveys, to measure the excess kurtosis $h_4$ of the stellar velocity
distribution, the latter parametrised as a Gauss-Hermite series. We find that
at all redshifts where we have large enough samples, $h_4$ anti-correlates with
the ratio between rotation and dispersion, highlighting the physical connection
between these two kinematic observables. In addition, and independently from
the anti-correlation with rotation-to-dispersion ratio, we also find a
correlation between $h_4$ and $M_\star$, potentially connected to the assembly
history of galaxies. In contrast, after controlling for mass, we find no
evidence of independent correlation between $h_4$ and aperture velocity
dispersion or galaxy size. These results hold for both star-forming and
quiescent galaxies. For quiescent galaxies, $h_4$ also correlates with
projected shape, even after controlling for the rotation-to-dispersion ratio.
At any given redshift, star-forming galaxies have lower $h_4$ compared to
quiescent galaxies, highlighting the link between kinematic structure and
star-forming activity.
