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The population of Be/X-ray binaries shows strong evidence of bimodality, especially in the spin period of neutron stars. Several physical mechanisms may produce this bimodality. The most favored candidate mechanisms are two distinct supernova channels or different accretion modes of the neutron stars in Be/X-ray binaries. Investigating the kinematics of these systems may provide some additional insight into the physics of this bimodality. If the two Be/X-ray binary subpopulations arise from two distinct supernova types, then the two subpopulations should have different peculiar (systemic) velocities. This can be tested either directly, by measuring the velocity of the system, or indirectly, by measuring the position of the system with respect to its birthplace. Using the most recent Gaia dataset and the newest catalogs of Small Magellanic Cloud (SMC) star clusters, we analyzed the tangential peculiar velocities of Be/X-ray binaries in the Galaxy and the positions of Be/X-ray binaries in the SMC. We used the distance of the system from the nearest young star cluster as a proxy to the tangential velocity of the system. We applied statistical testing to investigate whether the two subpopulations that are divided by the spin of the neutron star are also kinematically distinct. There is evidence that the two subpopulations are indeed kinematically distinct. However, the tangential peculiar velocities of the two subpopulations are the reverse from what is expected from the distinct supernova channel hypothesis. We find some marginal evidence ($p \approx$~0.005) that the Galactic Be/X-ray binaries from the short-spin subpopulation have systematically higher peculiar velocities than the systems from the long-spin subpopulation. The same effect, but weaker, is also recovered for the SMC Be/X-ray binaries for all considered cluster catalogs.