I don't know for sure, but I would guess that the war in Vietnam may have contributed to tearing down that belief in duty, at least in the United States. In Europe, perhaps it started its decline at the Nuremberg trials.
I think this confuses "Duty" with nationalism, it's peoples Duty to stop an unjust war. The Boy Scout oath says "On my honor I will do my best to do my duty to God and my country". You could replace God with any moral belief system.
I think our sense of Duty started to dissolve sometime in the 80s with the idea that "Greed is good" and then continued in the 90s with a rise in apathy. Both of which are selfish ideas, selfishness is the opposite of Duty.