The early 1930s was probably the peak of Jack Dragna's control over LA. Dragna was given a Commission seat. John Roselli took him to NY and vouched for him and his family to be recognized. But yeah, once Siegel moved west, New York clearly disregarded Dragna and gave Siegel the run of the place.

After Siegel died, Tommy Lucchese, who was Dragna's in-law, backed Dragna over Mickey Cohen. This is probably why so many Italians in Cohen's crew started switching to the Dragnas. The writing was on the wall. Cohen was one tenacious bastard, though. He took over all of Siegel's gambling rackets, didn't give the Dragnas a dime, and survived several attempts on his life.

Frattiano made the Dragna family seem like Murder Inc. in his book. Despite his proclamations of LA's power during his time, there just isn't a lot of court records or testimony from other turncoats to back it up. I don't believe his stories about pushing Moe Sedway around or Siegel speaking with him as equals for one second. Other than keeping up appearances with Dragna, Siegel wouldn't have wasted one second with Italians west of Chicago.

When the Milanos of Cleveland took over LA in the 80s they were still making some moves. A lot of guys in the LA family at that time were former Vegas guys who retreated to SoCal after the corporations bought the old Teamster casinos, and just about reaching retirement age. That was probably the last hurrah of the LA family. As others have said, New York guys operate in LA on their own now.