The operation, which began on October 10, is expected to take a month.
Bureau contractors hope to remove close to 2000 horses, but some will be released back to the rangeland after treatment with a long-term contraceptive.
The bureau has reported two deaths among the 702 so far brought into holding pens.
The first, on October 11, was put down to a combination of age and strain. The second, on October 15, was put down by a veterinarian because of "strain related to the gather".
The muster has been condemned by wild horse advocates.
The Cloud Foundation says the operation is estimated to cost at least $US3.5 million "and will leave the celebrated Adobe Town/Salt Wells Wild Horse Range - already populated with oil and gas wells - nearly devoid of mustangs".
Carol Walker, author/photographer of "Wild Hoofbeats: America's Vanishing Wild Horses", has been documenting the mustangs of Wyoming's Red Desert for seven years.
"I am witnessing the end of the wild West here as family after family of wild horses are driven into the trap, the majority of which will never return," she said.
"These are magnificent animals who deserve to live their lives in freedom with their families. They should not be stockpiled with the more than 40,000 wild horses and burros now in government holding."