We present the first results from the KMOS (K-band Multi-Object Spectrograph) AGN (active galactic nuclei) Survey at High redshift (KASHz), a VLT/KMOS integral-field spectroscopic (IFS) survey of z ≲ 0.6 AGN. We present galaxy-integrated spectra of 89 X-ray AGN (L_{2-10 keV} = 10⁴²-10⁴⁵ erg s^-1), for which we observed [OIII] (z ≈ 1.1-1.7) or Hα emission (z ≈ 0.6-1.1). The targets have X-ray luminosities representative of the parent AGN population and we explore the emission-line luminosities as a function of X-ray luminosity. For the [OIII] targets,≈50 per cent have ionized gas velocities indicative of gas that is dominated by outflows and/or highly turbulent material (i.e. overall linewidths≲600 km s^-1). The most luminous half (i.e. L_X > 6 × 10⁴³ erg s^-1) have a ≲2 times higher incidence of such velocities. On the basis of our results, we find no evidence that X-ray obscured AGN are more likely to host extreme kinematics than unobscured AGN. Our KASHz sample has a distribution of gas velocities that is consistent with a luminosity-matched sample ofz