Ahh yeah - .36 Navy below the 1860. In "Firearms of the American West 1866-1894, the authors talk of the soldiers (and civilians) "running buffalo" & shooting them from horseback for camp/fort food.
The Walkers that survived or had steel cylinders installed, were preferred, however the 1860 also worked. I suspect they were using conical loads, not round ball, but the Walker or Dragoons with round balls shooting 50gr. of power or more likely would work.
"It" was stated, that one of these guns noted above, held 6 buffalo in it.
They also stated they could not kill a buffalo with a .36.
On the other hand, the older .54 Martial pistols were more decisive.
The civilians were more likely to use plains-style rifles, SxS 12 bores, dueling or other pistols, loaded up a bit higher than normal.