Using integral field spectroscopy we investigate the kinematic properties of 35 massive centrally-dense and compact star-forming galaxies (${log{overline{M}_*}}=11.1$, $log{(Sigma_mathrm{1kpc})}>9.5$, $log{(M_ast/r_e^{1.5})}>10.3$) at $zsim0.7-3.7$ within the KMOS$^mathrm{3D}$survey. We spatially resolve 23 compact star-forming galaxies (SFGs) and find that the majority are dominated by rotational motions with velocities ranging from {$95-500$ km s$^{-1}$}. The range of rotation velocities is reflected in a similar range of integrated H$alpha$ linewidths, $75-400$ km s$^{-1}$, consistent with the kinematic properties of mass-matched extended galaxies from the full KMOS$^mathrm{3D}$ sample. The fraction of compact SFGs that are classified as `rotation-dominated or `disk-like also mirrors the fractions of the full KMOS$^mathrm{3D}$ sample. We show that integrated line-of-sight gas velocity dispersions from KMOS$^mathrm{3D}$ are best approximated by a linear combination of their rotation and turbulent velocities with a lesser but still significant contribution from galactic scale winds. The H$alpha$ exponential disk sizes of compact SFGs are on average $2.5pm0.2$ kpc, $1-2times$ the continuum sizes, in agreement with previous work. The compact SFGs have a $1.4times$ higher AGN incidence than the full KMOS$^mathrm{3D}$ sample at fixed stellar mass with average AGN fraction of 76%. Given their high and centrally concentrated stellar masses as well as stellar to dynamical mass ratios close to unity, the compact SFGs are likely to have low molecular gas fractions and to quench on a short time scale unless replenished with inflowing gas. The rotation in these compact systems suggests that their direct descendants are rotating passive galaxies.
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