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 similar to 2700 stars. In Paper IV we used these data to study the young stellar population. Here we study the kinematic properties through radial velocities (vr). The young stellar population remains kinematically associated with the molecular gas, following a similar to 10 km s(-1) gradient along the filament. However, near the center of the region, the vr distribution is slightly blueshifted and asymmetric;...