What are they going to do, call on Star Trek for the ships? It’s not like you can call a cab to go light years.
we theoretically have the technology for generational ships, and the future should make things a bit faster. we could use nuclear rockets, ion drives or even the original Project Orion which used nuclear bomb blasts to increase speed exponentially.
approaching something near the SOL is eventually possible but that, currently, is the speed limit, unless some of the theories of taking short cuts through worm holes, warp bubbles and the like. but this may prove to be a fantasy.
we can still reach out to the closer stars with generational ships......we may move out into the stars and lose track of one another, the distances and time being so vast........but if we set up viable colonies elsewhere, we move closer to preventing the human race from its eventual extinct....
Brave new world...