© 2017. The American Astronomical Society. All rights reserved. We present the first results of the K-band Multi-Object Spectrometer (KMOS) Lens-Amplified Spectroscopic Survey, a new ESO Very Large Telescope large program, doing multi-object integral field spectroscopy of galaxies gravitationally lensed behind seven galaxy clusters selected from the Hubble Space Telescope (HST) Grism Lens-Amplified Survey from Space. Using the power of the cluster magnification, we are able to reveal the kinematic structure of 25 galaxies at 0.7 z 2.3, in four cluster fields, with stellar masses 7.8log;(M∗/Mo) 10.5. This sample includes five sources at Z > 1 with lower stellar masses than in any previous kinematic integral field unit (IFU) surveys. Our sample displays a diversity in kinematic structure over this mass and redshift range. The majority of our kinematically resolved sample is rotationally supported, but with a lower ratio of rotational velocity to velocity dispersion than in the local universe, indicating the fraction of dynamically hot disks changes with cosmic time. We find that no galaxies with stellar mass <3 × 109M⊙ in our sample display regular ordered rotation. Using the enhanced spatial resolution from lensing, we resolve a lower number of dispersion-dominated systems compared to field surveys, competitive with findings from surveys using adaptive optics. We find that the KMOS IFUs recover emission line flux from HST grism-selected objects more faithfully than slit spectrographs. With artificial slits, we estimate that slit spectrographs miss, on average, 60% of the total flux of emission lines, which decreases rapidly if the emission line is spatially offset from the continuum.