Central galaxies (CGs) in massive halos live in unique environments with formation histories closely linked to that of the host halo. In local clusters, they have larger sizes (R-e) and lower velocity dispersions (sigma) at fixed stellar mass M-*, and much larger R-e at a fixed sigma than field and satellite galaxies (non-CGs). Using spectroscopic observations of group galaxies selected from the COSMOS survey, we compare the dynamical scaling relations of early-type CGs and non-CGs at z similar to 0.6 to distinguish possible mechanisms that produce the required evolution. CGs are systematically offset toward larger R-e at fixed sigma compared to non-CGs with similar M-*. The CG R-e-M-* relation also shows differences, primarily driven by a subpopulation (similar to 15%) of galaxies with large R-e, while the M-*-sigma relations are indistinguishable. These results are accentuated when double Sersic profiles, which better fit light in the outer regions of galaxies, are adopted. They suggest that even group-scale CGs can develop extended components by these redshifts that can increase total R-e and M-* estimates by factors of similar to 2. To probe the evolutionary link between our sample and cluster CGs, we also analyze two cluster samples at z similar to 0.6 and z similar to 0. We find similar results for the more massive halos at comparable z, but much more distinct CG scaling relations at low-z. Thus, the rapid, late-time accretion of outer components, perhaps via the stripping and accretion of satellites, would appear to be a key feature that distinguishes the evolutionary history of CGs.