That's a darn good question. I'd likely give the credit to either Bad Brains or Black Flag, since they're some of the pioneers in the genre, and they liked to borrow styles from other genres. Our Band Could Be Your Life: Scenes from the American Indie Underground is a decent book on their era if you're interested in reading more.

Black Flag used some jazz influences, and breakdowns were ─ if not common, then certainly not rare in that form. A Night in Tunisia is one big example that I can think of, and that dates from 1942.

Bad Brains borrowed from damn near everything, it seems like, and the album I Against I might contain the first examples of what you're looking for. That was released in 1986.

You're going to have to take what I say with a grain of salt, though. I'm not a specialist in musical history, and I'm definitely not a metalcore aficionado.