(euphemistic) To die; to follow a course leading to death or extinction.
1888, Thomas Hardy, "The Three Strangers," in Wessex Tales,
The first stranger handed his neighbor the family mug—a huge vessel of brown ware, having its upper edge worn away like a threshold by the rub of whole generations of thirsty lips that had gone the way of all flesh.
On the U.S. West Coast, the clubs and restaurants that feature topless female entertainers and waitresses also seemed to be going the way of all flesh. In Los Angeles, 20% of the joints have closed.
2006, Laura Wertheimer, "Clerical Dissent, Popular Piety, and Sanctity in 14th-Century Peterborough," Journal of British Studies, vol. 45, no. 1, p. 3,
Laurence of Oxford went the way of all flesh on the gallows.