Here's a rather captivating artist's conception of a lunar avalanche. The artist is sadly unknown, but Chicago's Museum of Science and Industry displayed the piece in 1960 to illustrate the future of manned moon exploration.
The first humans didn't arrive on the lunar surface till 1969, and our species last visited in 1972. Interestingly enough, that final mission did feature an avalanche site. Apollo 17 touched down int he lunar valley Taurus-Littrow, which contained ancient avalanche deposits. Thankfully, the episode was not as action-packed as the illustration above.
Naturally, avalanches do occur on the moon -- they occur anywhere there's sufficient gravity, material and flux. Saturn's moon Iapetus boasts massive ice avalanches and you should see the ones they have on Mars.