Gamma-ray emission in radio galaxies under the VLBI scope -- II. The relationship between gamma-ray emission and parsec-scale jets in radio galaxies.

Following our study of the radio and high-energy properties of $\gamma$-ray-emitting radio galaxies, here we investigate the kinematic and spectral properties of the parsec-scale jets of radio galaxies that have not yet been detected by Fermi-LAT. We take advantage of the regular VLBI observations provided by the TANAMI monitoring program, and explore the kinematic properties of six $\gamma$-ray-faint radio galaxies. We include publicly available VLBI kinematics of $\gamma$-ray-quiet radio galaxies monitored by the MOJAVE program and perform a Fermi-LAT analysis, deriving upper limits. We combine these results with those from our previous paper to construct the largest sample of radio galaxies with combined VLBI and $\gamma$-ray measurements to date. We find superluminal motion up to $\beta_\mathrm{app}=3.6$ in the jet of PKS 2153$-$69. We find a clear trend of higher apparent speed as a function of distance from the jet core on scales of $\sim10^5\,R_s$, corresponding to the end of the collimation and acceleration zone in nearby radio galaxies. We find evidence of subluminal apparent motion in the jets of PKS 1258$-$321 and IC 4296, and no measurable motion for PKS 1549$-$79, PKS 1733$-$565 and PKS 2027$-$308. We compare the VLBI properties of $\gamma$-ray-detected and undetected radio galaxies, and find significantly different distributions of median core flux density, and, possibly, of median core brightness temperature. We find a significant correlation between median core flux density and $\gamma$-ray flux, but no correlation with typical Doppler boosting indicators such as median core brightness temperature and core dominance. Our study suggests that high-energy emission from radio galaxies is related to parsec-scale radio emission from the inner jet, but is not driven by Doppler boosting effects, in contrast to the situation in their blazar counterparts.