Let's first put aside the use of "correspondence" to mean "connection," as in "the correspondences between German and English prove that English is a Teutonic language at heart." In that sense, the plural with "s" is necessary and proper.

Turning to "correspondence" in the sense of "one or more exchanges of written communication," I think the use of the plural would be rare, but when needed, justified. "The Boston Tea Party led to the famous correspondence between Washington and Jefferson, but many such correspondences sprang up between pairs of famous men in the wake of that notorious event."

Maybe we can say that one uses the plural only to emphasize the particularity of a set of exchanges.

"We had a heated correspondence for several months in 2000-2001 about Bush's election (or Gore's election, if you will), and another, briefer correspondence in 2004-2005, after Kerry went down in flames. Both correspondences were published together in book form in 2006."