In this second installment of the series, we look at the internal kinematics of the multiple stellar populations of the globular cluster $omega$ Centauri in one of the parallel Hubble Space Telescope (HST) fields, located at about 3.5 half-light radii from the center of the cluster. Thanks to the over 15-year-long baseline and the exquisite astrometric precision of the HST cameras, well-measured stars in our proper-motion catalog have errors as low as $sim 10 mu$as yr$^{-1}$, and the catalog itself extends to near the hydrogen-burning limit of the cluster. We show that second-generation (2G) stars are significantly more radially anisotropic than first-generation (1G) stars. The latter are instead consistent with an isotropic velocity distribution. In addition, 1G have excess systemic rotation in the plane of the sky with respect to 2G stars. We show that the six populations below the main-sequence (MS) knee identified in our first paper are associated to the five main population groups recently isolated on the upper MS in the core of cluster. Furthermore, we find both 1G and 2G stars in the field to be far from being in energy equipartition, with $eta_{rm 1G}=-0.007pm0.026$ for the former, and $eta_{rm 2G}=0.074pm0.029$ for the latter, where $eta$ is defined so that the velocity dispersion $sigma_mu$ scales with stellar mass as $sigma_mu propto m^{-eta}$. The kinematical differences reported here can help constrain the formation mechanisms for the multiple stellar populations in $omega$ Centauri and other globular clusters. We make our astro-photometric catalog publicly available.
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