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@Angst has correctly traced this back to New Orleans (thanks for doing the underlying research).However, it is nearly impossible to search music just by "New Orleans" alone, since New Orleans has such a rich, diverse and full musical history, and is arguably the birthplace of what we know as "jazz."

I'd suggest "jazz dirge," "blues dirge," or "New Orleans dirge" as the genre instead. Traditionally, "dirge" (a slow, mournful song played on the way to the graveyard) and "Second Line" (an uptempo, joyful piece played post interment, with crowd participation) are the parts of a New Orleans jazz funeral.

New Orleans jazz legend Wynton Marsalis' "Majesty of the Blues" is a great example of a jazz/blues album deliberately patterned in this way. Tom Waits' "Anywhere I Lay My Head" has both parts in a single song.