Mike Barthel traces the beginnings to Jamaica in the 1960s:

The island's extremely strong local music culture enabled a tight interplay between the people who made records and the people who listened and danced to them. As DJ/producers like Lee "Scratch" Perry saw that crowds were interested in longer musical experiences than they could provide with the single format (the 7" vinyl disc on which singles came was generally only able to hold four minutes' worth of music without suffering a drop in sound quality), they produced new versions which they referred to as "versions." … Since Jamaican producers tended to be owner-operators, both recording and distributing their own music, they had access to the multi-track masters at a time when American musicians didn't. In the United States (and other industrialized countries), the master recordings were owned by the record company, making it much more difficult for artists to get their hands on their own recordings, let alone legally release remixes.

(Video: a remix of Yelle’s "Comme un Enfant," starring Nathan Barnatt)