1Ulsan National Institute of Science and Technology 2Los Alamos National Laboratory, USA 3Pusan National University Radio galaxy jets emitted from the vicinity of central AGN reach ultra-relativistic speeds within a few parsecs, but depending on the properties of the jet, the ambient media profile of the host galaxy, and the rate of mass-loading from the stellar winds, the jets evolve differently. The evolutionary outlook of such jets (Kunert-Bajrawzewska et al. 2005) suggests that, depending on the temporal stage of evolution, these jets are observed as Compact Symmetric Objects (CSO), Middle-sized Symmetric Objects (MSO) and Large Symmetric Objects (LSO, e.g., Fanaroff-Riley or FR sources). The core radius of host galaxies, beyond which ambient pressure falls sharply, is close to the transition size of the CSO and MSO. Observations show that the size of the hot spot and the linear size of the radio source exhibit a broken power-law correlation, where the slope changes around a similar length scale (Kawakatu et al. 2008). It has been suggested that the change in slope of the ambient density/pressure distribution and the jet-head properties at the core radius determine the track of evolution (Kawakatu et al. 2009). However, a multi-variate study on the early stages of radio galaxy jets, especially on the CSO-MSO-FR connection, is yet to be done within the framework 3D relativistic jet simulations. We expand our existing work on FR-I jets (Bhattacharjee et al., 2024) and study the flow structures and morphological properties of CSOs and MSOs. We also study the energetics and dynamics of the jet, especially the jet head, to obtain the propagation rate of the brightest region (or the 'hot spot'), as it evolves through stratified ambient media, being modified by shocks, expansions, and mass-loading, which has not been done before. Furthermore, we also study the acceleration/deceleration of jets across the galactic core by varying jet power and ambient slope and connect them with the CSO-MSO transition features seen in observation.