  Theory predicts that we should find fast, ejected (runaway) stars of all
masses around dense, young star-forming regions. $N$-body simulations show that
the number and distribution of these ejected stars could be used to constrain
the initial spatial and kinematic substructure of the regions. We search for
runaway and slower walkaway stars within 100 pc of the Orion Nebula Cluster
(ONC) using $Gaia$ DR2 astrometry and photometry. We compare our findings to
predictions for the number and velocity distributions of runaway stars from
simulations that we run for 4 Myr with initial conditions tailored to the ONC.
In $Gaia$ DR2, we find 31 runaway and 54 walkaway candidates based on proper
motion, but not all of these are viable candidates in three dimensions. About
40 per cent are missing radial velocities, but we can trace back 9 3D-runaways
and 24 3D-walkaways to the ONC, all of which are low/intermediate-mass (<8
M$_{\odot}$). Our simulations show that the number of runaways within 100 pc
decreases the older a region is (as they quickly travel beyond this boundary),
whereas the number of walkaways increases up to 3 Myr. We find fewer walkaways
in $Gaia$ DR2 than the maximum suggested from our simulations, which may be due
to observational incompleteness. However, the number of $Gaia$ DR2 runaways
agrees with the number from our simulations during an age of $\sim$1.3-2.4 Myr,
allowing us to confirm existing age estimates for the ONC (and potentially
other star-forming regions) using runaway stars.
