Chogir (K2) and Shaksgam Rivers, Xinjiang, 1996
These river segments were run by Vladimir Lysenko and Boris Ivanov of the the Himalayan Club of Rafters and Kayakers (http://www.gkrkr.web-online.ru/). They drain the north (Chinese) side of K2, one of the highest mountains on earth, and probably the most difficult to climb. Due to the extreme desert environment, there were no inhabitants along the rivers.
They put-in at about 17,000' near 36.411303°, 76.078949° and ran about 60 miles in 7 days to the Yarkant, then repeated the run on the Yarkant that Sobek had done in 1990. The average gradient was about 60' per mile and the river was Class 3-4 at the flow of about 3000 cfs near the confluence with the Yarkant (36.395054°, 76.685209°).
See Rafting Down The Highest World Peaks by Vladimir Lysenko for details. This journal also describes their first descent of the Rong Chu, a tributary to the Phung Chu which drains the Tibetan side of Mt. Everest.