This study constitutes part of a larger effort aimed at better characterizing the Galactic globular clusters (GGCs) located towards the inner Milky Way bulge and disk. Here, we focus on internal kinematics of nine GGCs, obtained from space-based imaging over time baselines of $>$9 years. We exploit multiple avenues to assess the dynamical state of the target GGCs, constructing radial profiles of projected stellar density, proper motion dispersion, and anisotropy. We posit that two-thirds (6/9) of our target GGCs are in an advanced dynamical state, and are close to (or have recently undergone) core collapse, supported by at least two lines of evidence: First, we find relatively steep proper motion dispersion profiles, in accord with literature values for core-collapsed GGCs. Second, we find that our sample is, in the mean, isotropic even out to their half-light radii, although one of our target clusters (NGC 6380) is tangentially anisotropic at $>$1$sigma$ beyond its half-light radius, in accord with theoretical predictions for clusters evolving in strong tidal fields. Our proper motion dispersion and anisotropy profiles are made publicly available.
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