That's not what happens though. Within a few hundred years the beings living on the other planet would likely not be considered human due to significant changes in physiology based on new gravity, atmospheric pressure and density, atmospheric composition, differences in star output and radiation. Their culture would change immediately, because none of the places or items available on earth would be there. Also, if it's embryos, they do not have culture. It would have to be forced on them by the robots.

The sun is well understood and will be safe for 1 billion years at least, at which point it's impossible we would still be the same species we are today unless humanity pursues genomics and stops all genetic mutations permanently.

If we explore the galaxy, it's most likely to be done by small probes that can self-replicate and therefore bounce from planet to planet instead of requiring new machines be sent from earth. The time scales are too large, the distances too great, and the supposed benefits are not felt or experienced by anyone.