We have measured the absolute proper motions of nine low-latitude, inner Galaxy globular clusters, namely, NGC?6273 (M 19), NGC?6284, NGC?6287, NGC?6293, NGC?6333 (M 9), NGC?6342, NGC?6356, NGC?6388, and NGC?6441. These are the first determinations ever made for these clusters. The proper motions are on the International Celestial Reference System via Hipparcos. The proper-motion errors range between 0.4 and 0.9 mas yr–1 and are dominated by the number of measurable cluster members in these regions which are very crowded by the bulge/bar and the thick disk. This sample contains five metal-poor ([Fe/H] –1.0) and four metal-rich ([Fe/H] ≥ –1.0) clusters; seven clusters are located within ~4 kpc from the Galactic center, while the remaining two, namely NGC?6356 and NGC?6284, are in the background of the bulge at ~7.5 kpc from the Galactic center. By combining proper motions with radial velocities and distances from the literature we derive three-dimensional velocities. In a number of cases, distance uncertainties make the kinematical classification ambiguous. For the metal-poor group of clusters, we find that three clusters, namely NGC?6273, NGC?6287, and NGC?6293 are members of a kinematically hot system, the inner halo. As for the remaining two metal-poor clusters, NGC?6284 is located at ~7.5 kpc from the Galactic center and kinematically belongs to the thick disk, while NGC?6333, located in the inner ~2 kpc, has an uncertain membership (between halo and thick disk) due to the distance uncertainty. Within the metal-rich group of clusters, NGC?6356 and NGC?6342 have velocities compatible with membership in the thick disk; however, velocity uncertainties do not allow us to rule out their belonging to a hotter system. NGC?6342 is within ~2 kpc from the Galactic center, and thus it may belong to the bulge. NGC?6356 is at ~7.5 kpc from the Galactic center and thus its metallicity, kinematics, and location argue together in favor of thick-disk membership. The remaining two metal-rich clusters, NGC?6388 and NGC?6441, have velocities incompatible with membership in the thick disk or the bar of the Milky Way. They can be thought of as members of a kinematically hot system in the inner Galaxy. Curiously, both clusters have similar velocity components. Together with their similar Galactic location and peculiar but similar stellar-population characteristics, these two clusters may share a common origin. Their velocities are also very low indicating that the two clusters are now at apocenter, i.e., they will not leave the inner ~4 kpc of the Galaxy.
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