Context. 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. Aims. 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. Methods. We calculated proper motions based on multi-epoch Ks-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. Results. We obtained proper motions for a total of 576 411 unique sources over an area of 23 deg2 covering the Magellanic Bridge including mainly Milky Way foreground stars, background galaxies, and a small population of possible Magellanic Bridge stars (< 15 000), which mostly consist of giant stars with 11.0 < Ks < 19.5 mag. The first proper motion measurement of the Magellanic Bridge centre is 1.80 ± 0.25 mas yr-1 in right ascension and-0.72 ± 0.13 mas yr-1 in declination. The proper motion measurements of stars along the Magellanic Bridge from the VISTA survey of the Magellanic Cloud system (VMC) and Gaia DR2 data 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. Conclusions. Our measurements indicate that the Magellanic Bridge is stretching. By converting the proper motions to tangential velocities, we obtain ∼110 km s-1 in the plane of the sky. Therefore it would take a star roughly 177 Myr to cross the Magellanic Bridge.