  We present GaiaHub, a publicly available tool that combines $Gaia$
measurements with $Hubble$ $Space$ $Telescope$ ($HST$) archival images to
derive proper motions (PMs). It increases the scientific impact of both
observatories beyond their individual capabilities. $Gaia$ provides PMs across
the whole sky, but the limited mirror size and time baseline restrict the best
PM performance to relatively bright stars. $HST$ can measure accurate PMs for
much fainter stars over a small field, but this requires two epochs of
observation which are not always available. GaiaHub yields considerably
improved PM accuracy compared to $Gaia$-only measurements, especially for faint
sources $(G \gtrsim 18)$, requiring only a single epoch of $HST$ data observed
more than $\sim 7$ years ago (before 2012). This provides considerable
scientific value especially for dynamical studies of stellar systems or
structures in and beyond the Milky Way (MW) halo, for which the member stars
are generally faint. To illustrate the capabilities and demonstrate the
accuracy of GaiaHub, we apply it to samples of MW globular clusters (GCs) and
classical dwarf spheroidal (dSph) satellite galaxies. This allows us, e.g., to
measure the velocity dispersions in the plane of the sky for objects out to and
beyond $\sim 100$ kpc. We find, on average, mild radial velocity anisotropy in
GCs, consistent with existing results for more nearby samples. We observe a
correlation between the internal kinematics of the clusters and their
ellipticity, with more isotropic clusters being, on average, more round. Our
results also support previous findings that Draco and Sculptor dSph galaxies
appear to be radially anisotropic systems.
