The phrase has an old-fashioned sound, suitable for historical or literary contexts. In his novel, the author described the knight as having gone the way of all flesh.
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Metaphorical Use
It is often used metaphorically to imply the end of non-living things like traditions or technology. With the rise of e-books, it's feared that libraries may go the way of all flesh.
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Irrelevance
It suggests not just death or decay but also becoming outdated or no longer useful. As streaming services gain popularity, DVDs are going the way of all flesh.