10,000 years is not a large amount time for the Milky Way, but a very long time for a civilization. In that amount of time, humanity could find its self in another dark ages. Would they still be in contact with the people who colonized another star after that amount of time, even if society collapsed into a dark age? If the two groups were no longer in contact, would they still be considered the same civilization? Also isn’t the galactic core a very dangerous place for leaving things because of density of stars and frequent solar flares? From what I know, life is more likely to survive in the galactic arms, where there is less density of stars and thus less solar flares to wipe out all living things.