  The Magellanic Clouds are a nearby pair of interacting dwarf galaxies and
satellites of the Milky Way. Studying their kinematic properties is essential
to understanding their origin and dynamical evolution. They have prominent
tidal features and the kinematics of these features can give hints about the
formation of tidal dwarfs, galaxy merging and the stripping of gas. In addition
they are an example of dwarf galaxies that are in the process of merging with a
massive galaxy. The goal of this study is to investigate the kinematics of the
Magellanic Bridge, a tidal feature connecting the Magellanic Clouds, using
stellar proper motions to understand their most recent interaction. We
calculated proper motions based on multi-epoch $K_{s}$-band aperture
photometry, which were obtained with the Visible and Infrared Survey Telescope
for Astronomy (VISTA), spanning a time of 1-3 yr, and we compared them with
$Gaia$ Data Release 2 (DR2) proper motions. We tested two methods for removing
Milky Way foreground stars using $Gaia$~DR2 parallaxes in combination with
VISTA photometry or using distances based on Bayesian inference. We obtained
proper motions for a total of 576,411 unique sources over an area of $23$
deg$^{2}$ covering the Magellanic Bridge including mainly Milky Way foreground
stars, background galaxies, and a small population of possible Magellanic
Bridge stars ($<$15,000). The first proper motion measurement of the Magellanic
Bridge centre is $1.80\pm0.25$ mas yr$^{-1}$ in right ascension and
$-0.72\pm0.13$ mas yr$^{-1}$ in declination. The proper motion measurements
confirm a flow motion from the Small to the Large Magellanic Cloud. This flow
can now be measured all across the entire length of the Magellanic Bridge. Our
measurements indicate that the Magellanic Bridge is stretching.
