'Mushed' meaning
Mush is the traditional cry for encouraging sled dogs—“Mush, you huskies!”
Etymonline conjectures
first recorded 1862, as mouche, perhaps altered from French marchons! "advance!" (imperative of marcher "to march)"
You are right, it seems to mean 'just going', as the AHD notes.
mush 2 (mŭsh)
v. mushed, mush·ing, mush·es
v.intr.
To travel, especially over snow with a dogsled.