A question about the Drug Pipeline

It seems that all the Families were involved in drugs during the 30s and 40s but seemingly from different channels (before it really got organized at the Havana Conference and furthermore during the 1957 summit held by Bonanno and Luciano.

Is it safe to say, it started under the control of the Lucianos in 30s/40s, then it shifted to the Bonannos in the 50s/60s ,eventually controlled by the Luccheses and ultimately the Gambinos 70s/80s?

"Joe Bananas went after Carlo Gambino, the war went on for seven years..... When guys go to the mattresses, they're not out earning" -Tony Soprano