There wasnt exatly a "smooth transition" after Albert was murdered though. There was a war on the verge of breaking out, at least two guys were murdered as a result of this "almost war", one of them being Armand "Tommy" Rava. Anastasia had his loyalists, among them Neil Dellacroce. Rava was scheming to takeover the family, and didn't like the fact that Carlo basically weaseled his way to the boss seat, by conspiring with members from other families to kill Anastasia. The Anastasia loyalists, which were headed by Rava & Dellacroce actually opposed the Gambino/Biondo group that moved to take power after Anastasias death. Things only came to an end after Gambino loyalists killed Armand Rava, and Gambino compensated the rest of the Anastasia loyalists by making Dellacroce his underboss.

Even during the days of John Gotti, there were still members of the Gambino Family whom could trace their origins back to Albert Anastasia and his initial crew.