  Runaway stars are ejected from their place of birth in the Galactic disk,
with some young B-type runaways found several tens of kiloparsecs from the
plane traveling at speeds beyond the escape velocity. Young open clusters are a
likely place of origin, and ejection may be either through N-body interactions
or in binary supernova explosions. The excellent quality of Gaia astrometry
opens up the path to study the kinematics of young runaway stars to such a high
precision that the place of origin in open stellar clusters can be identified
uniquely. We developed an efficient minimization method to calculate whether
two or more objects may come from the same place, which we tested against
samples of Orion runaways. Our fitting procedure was then used to calculate
trajectories for known runaway stars where we used Gaia data and updated radial
velocities. We found that only half of the sample could be classified as
runaways while the others were walkaway stars. Most of the latter stars turned
out to be binaries. We identified parent clusters for runaways based on their
trajectories and then used cluster age and flight time of the stars to
investigate whether the ejection was likely due to a binary supernova or due to
a dynamical ejection. In particular, we show that the classical runaways AE
Aurigae and $\mu$ Columbae might not have originated together, with $\mu$
Columbae having an earlier ejection from Collinder 69, a cluster near the ONC.
The second sample investigated comprises a set of distant runaway B stars in
the halo which have been studied carefully by quantitative spectral analyses.
We are able to identify candidate parent clusters for at least four stars
including the hyper-runaway candidate HIP 60350. The ejection events had to be
very violent, ejecting stars at velocities as large as 150 to 400 km/s.
