Ideally we would send artificially intelligent machines who don’t mind a journey that takes hundreds of years to explore and begin building infrastructure thousands of years before any humans will arrive. Then when the star is at its closest there is a mad dash of humans to populate the system.
Do this every time a star is going to approach to within 4-5 LY and then from those stars when they approach other stars, and so on. It starts to seem feasible for the human race to overcome the vast engineering challenges of spreading throughout the galaxy. It would be difficult to get humans to put effort into something that takes more than a lifetime to accomplish though.