If you take real physics, you could travel to the next star in moving cities.
Imagine independent, self-sufficient mining outposts attached to an asteroid. While the asteroid is being used up, the youth will start to search the next target. People will follow when one is found. Latest when the last scrap of material is used, also the most stagnant family will move on.
You can live that way, in a few generations, from asteroid to asteroid to Jupiter's trojans, to Saturn's rings, into the plutonoids, from there into the Oort cloud.... well and at the end of our Oort cloud, the next star's gravity well begins. From snowball to snowball, in a few 100 years you arrive at Proxima Centauri.
It is doable with today's technology or very little advancement from here. But you have to say goodbye to the dream of going there, taking some photos and coming back to show them your wife.
Instead you have to develop an entirely self sufficient life style which is religiously or philosophically focused on moving on to the next snowball, you have to take your entire family with you and you have to say goodbye to earth.
What tech would be needed that is absolutely not available today? For the sake of this question, I would understand rotating hammers or rings for artificial gravity as existing technology: even if it has not yet been built, the concept is understood and technically feasible.
Which prerequisites are missing today?
What problems could those space faring families encounter?