A "political dynasty" refers to a situation where members of the same family continuously hold influential positions in the government, resulting in a concentration of power within that family. Full definition
I grew up admiring Pierre Trudeau (not fully realizing how much he and I differed, politically, from my Albertan grandparents), and even though I'm not a fan of political dynasties, per se, I definitely admire and support your new PM - designate. (highheelsinthewilderness.com)
As The Journal's Michael Howard Saul reported, Democratic nominee Andrew Cuomo, the current state attorney general and scion of famous political dynasty in New York, has given up much of his lead in recent opinion polls. (blogs.wsj.com)
McLaughlin was asked about Albany's troubled history with political dynasties and explained her support for term limits: (thealt.com)