  Various stellar objects experience a velocity kick at some point in their
evolution. These include neutron stars and black holes at their birth or binary
systems when one of the two components goes supernova. For most of these
objects, the magnitude of the kick and its impact on the object dynamics
remains a topic of debate. We investigate how kicks alter the velocity
distribution of objects born in the Milky Way disc, both immediately after the
kick and at later times, and whether these kicks are encoded in the observed
population of Galactic neutron stars. We simulate the Galactic trajectories of
point masses on circular orbits in the disc after being perturbed by an
isotropic kick, with a Maxwellian distribution of magnitudes with $\sigma=265$
km/s. Then, we simulate the motion of these point masses for $200$ Myr. These
trajectories are then evaluated, either for the Milky Way population as a whole
or for those passing within two kiloparsecs of the Sun, to get the time
evolution of the velocities. During the first $20$ Myr, the bulk velocity of
kicked objects becomes temporarily aligned to the cylindrical radius, implying
an anisotropy in the velocity orientations. Beyond this age, the velocity
distribution shifts toward lower values and settles to a median of $\sim200$
km/s. Around the Sun, the distribution also loses its upper tail, primarily due
to unbound objects escaping the Galaxy. We compare this to the velocities of
Galactic pulsars and find that pulsars show a similar evolution with
characteristic age. The shift of the velocity distribution is due to bound
objects spending most of their orbits at larger radii after the kick. They are,
therefore, decelerated by the Galactic potential. We find the same deceleration
to be predicted for nearby objects and the total population and conclude it is
also observed in Galactic pulsars.
