I believe they are generally in trail runners. Perhaps others can chime in, this is just an anecdote:
We hiked up to Galehead (mountain and hut) a couple weeks ago, and I specifically recall pausing at a water crossing to allow a hut croo member to pass, with their load of trash on their way down. I watched them get a foot wet in the water, and saw runners, not boots.
They also still have those wooden external frame packs. I haven't ever tried one on, but they look miserable. I suppose it's tradition, and perhaps a good way to transport their heavy loads.