We analyse the velocity dispersion properties of 472 z \sim 0.9 star-forming galaxies observed as part of the KMOS Redshift One Spectroscopic Survey ( KROSS ) . The majority of this sample is rotationally dominated ( 83 \pm 5 % with v _ { C } / \sigma _ { 0 } > 1 ) but also dynamically hot and highly turbulent . After correcting for beam smearing effects , the median intrinsic velocity dispersion for the final sample is \sigma _ { 0 } = 43.2 \pm 0.8 km s ^ { -1 } with a rotational velocity to dispersion ratio of v _ { C } / \sigma _ { 0 } = 2.6 \pm 0.1 . To explore the relationship between velocity dispersion , stellar mass , star formation rate and redshift we combine KROSS with data from the SAMI survey ( z \sim 0.05 ) and an intermediate redshift MUSE sample ( z \sim 0.5 ) . While there is , at most , a weak trend between velocity dispersion and stellar mass , at fixed mass there is a strong increase with redshift . At all redshifts , galaxies appear to follow the same weak trend of increasing velocity dispersion with star formation rate . Our results are consistent with an evolution of galaxy dynamics driven by disks that are more gas rich , and increasingly gravitationally unstable , as a function of increasing redshift . Finally , we test two analytic models that predict turbulence is driven by either gravitational instabilities or stellar feedback . Both provide an adequate description of the data , and further observations are required to rule out either model .