I think there's several things. I think the widest war was set off by the killing of McCluskey. This was Vito's legacy - he had set up ten years before a system of conduct, and the Corleones breached that. The other families, who had previously been recruited into the system, were now required to set upon the Corleones until the Corleones fell back into line.

Second is the drugs business. The families were probably seeing freelancers make a lot of money, and realized both the threat and opportunity in that. To gain control over that business, the families needed to deploy their advantages over outsiders, and the Corleones controlled several of those functions. Again, the Corleones had to be brought into line.

Finally, I think there's also an opportunistic grab for power by Barzini. He'd be pushing in any direction, no matter the issue, with the goal of chipping away at the Corleones.