  The kinematics and dynamics of young stellar populations enable us to test
theories of star formation. With this aim, we continue our analysis of the
SDSS-III/APOGEE IN-SYNC survey, a high resolution near infrared spectroscopic
survey of young clusters. We focus on the Orion A star-forming region, for
which IN-SYNC obtained spectra of $\sim2700$ stars. In Paper IV we used these
data to study the young stellar population. Here we study the kinematic
properties through radial velocities ($v_r$). The young stellar population
remains kinematically associated with the molecular gas, following a
$\sim10\:{\rm{km\:s}}^{-1}$ gradient along filament. However, near the center
of the region, the $v_r$ distribution is slightly blueshifted and asymmetric;
we suggest that this population, which is older, is slightly in foreground. We
find evidence for kinematic subclustering, detecting statistically significant
groupings of co-located stars with coherent motions. These are mostly in the
lower-density regions of the cloud, while the ONC radial velocities are
smoothly distributed, consistent with it being an older, more dynamically
evolved cluster. The velocity dispersion $\sigma_v$ varies along the filament.
The ONC appears virialized, or just slightly supervirial, consistent with an
old dynamical age. Here there is also some evidence for on-going expansion,
from a $v_r$--extinction correlation. In the southern filament, $\sigma_v$ is
$\sim2$--$3$ times larger than virial in the L1641N region, where we infer a
superposition along the line of sight of stellar sub-populations, detached from
the gas. On the contrary, $\sigma_v$ decreases towards L1641S, where the
population is again in agreement with a virial state.
