Heinlein and others have shown that the poor bastards in a colony ship will arrive to find the planet already inhabited and thriving from the generations of colonists who hopped the FTL ship that left earth after the tech was invented a century later.
Don’t really need FTL. Build a ship that travels .5 SoL and they wouls reach Proxima in 8 or nine years . What gappens after tgat depends on what they find. If they find no hospitable planets maybe on their return they search out for the first ship and lead them home. Turtledove created a similar scenario in tge last book of the Worl War Series.