Well, there's the possible future kids of some heroes like Clark and Diana, but in continuity I think the closest DC has is the Knight family.
Which isn't very close at all, since the Summers are all very powerful mutants and the Knights are just very influential "normals."
But, the Knights, like the Summers, are basically DC's big, extended "royal" family. We've got Ted Knight, the first Starman and creator of cosmic energy technology (which in many futures saves the world from the energy crisis). We've got his....cousin?.....Sandra, the first Phantom Lady and Headmistress of the spy school that trains the Phantoms. We've got Ted's oldest son who was Starman for five seconds, Jack, his youngest son and probably the most popular Starman (at least during his time). There's Stormy Knight, the Phantom Lady of pre-52, and her....father?.....was President of the United States (until being killed and replaced with an android from Apokolips).
I believe the family is also related to Kate Spencer, the current Manhunter, and have marriage ties (I think?) to the Pratt family (the original Atom) and maybe Arn "Iron" Munroe (a Gladiator rip-off who replaced Superman in the Golden Age history of clutter earth).
There's also Knights from the future, including the Starman of DC 1 Million who started out a traitor and became a hero.
WAIT!
NO!
We have the Allen-West families (who seem to inter-marry so much across generations they're basically one clan). That gives us Barry, Bart, Wally, Wallace, the Tornado Twins, XS, and a ton of other Flashes from the future, as well as Wally's daughter Irey, who was the second Impulse for all of two seconds before the reboot. I dont think we have as many prevalent West-Allens in the modern day (reboot foolishness notwithstanding) as we have Knights, but the Flash dynasty are among the most powerful people on earth and are always involved in the big events with the League, so they're a better parallel to the Summers than the Knights.
But for real, the Knights are the Kennedy's of the DCU.