We explore the hypothesis that B type emission-line stars (Be stars) have their origin in mass-transfer binaries by measuring the fraction of runaway Be stars. We assemble the largest-to-date catalogue of 632 Be stars with 6D kinematics, exploiting the precise astrometry of the Tycho-Gaia Astrometric Solution (TGAS) from the first Gaia Data Release. Using binary stellar evolution simulations, we make predictions for the runaway and equatorial rotation velocities of a runaway Be star population. Accounting for observational biases, we calculate that if all classical Be stars originated through mass transfer in binaries, then $17.5\%$ of the Be stars in our catalogue should be runaways. The remaining $82.5\%$ should be in binaries with subdwarfs, white dwarfs or neutron stars, because those systems either remained bound post-supernova or avoided the supernova entirely. Using a Bayesian methodology, we compare the hypothesis that each Be star in our catalogue is a runaway to the null hypothesis that it is a member of the Milky Way disc. We find that $13.1-2.4+2.6\%$ of the Be stars in our catalogue are runaways, and identify a subset of 40 high-probability runaways. We argue that deficiencies in our understanding of binary stellar evolution, as well as the degeneracy between velocity dispersion and number of runaway stars, can explain the slightly lower runaway fraction. We thus conclude that all Be stars could be explained by an origin in mass-transfer binaries. This conclusion is testable with the second Gaia data release (DR2).