In the framework of the TANAMI multi-wavelength and VLBI monitoring, we study the evolution of the parsec-scale radio emission in radio galaxies in the southern hemisphere and their relationship to the $gamma$-ray properties. In this first paper, we focus on Fermi-LAT-detected sources. We perform a kinematic analysis for five $gamma$-ray detected radio galaxies using multi-epoch 8.4 GHz VLBI images, deriving limits on intrinsic jet parameters. We analyzed Fermi-LAT data in order to study possible connections between the $gamma$-ray properties and the pc-scale jets of Fermi-LAT-detected radio galaxies, both in terms of variability and average properties. We discuss the individual source results and draw preliminary conclusions on sample properties including published VLBI results from the MOJAVE survey, with a total of fifteen sources. We find that the first $gamma$-ray detection of Pictor A might be associated with the passage of a new VLBI component through the radio core. For the peculiar AGN PKS 0521-36, we detect subluminal parsec-scale jet motions, and we confirm the presence of fast $gamma$-ray variability in the source down to timescales of 6 hours. We robustly confirm the presence of significant superluminal motion, up to $beta_{app}sim$3, in the jet of the TeV radio galaxy PKS 0625-35. Finally, we place a lower limit on the age of the Compact Symmetric Object (CSO) PKS 1718-649. We draw some preliminary conclusions on the relationship between pc-scale jets and $gamma$-ray emission in radio galaxies. We find that the VLBI core flux density correlates with the $gamma$-ray flux, as seen in blazars. On the other hand, the $gamma$-ray luminosity does not show any dependence on the core brightness temperature and core dominance, two indicators of Doppler boosting, suggesting that $gamma$-ray emission in radio galaxies is not driven by orientation-dependent effects.
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