Thought I'd share this response from the Indian Territory message board regarding these men:

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Ken,
I don't think it's correct to call these men deserters, as their term of enlistment in the 1st Cherokee ended on May 12, 1864, about two months before they turned up in Santa Fe. As citizens of Arkansas they were free to leave when their term expired, although they would obviously be subject to Confederate conscription laws which applied in the state of Arkansas. It seems more like a case of these men being fed up with the war and wanting to get away from it all. One way of doing so would be to go west and claim to be Confederate deserters and take the oath of allegiance. They would then be free to go about their own way without having to take further part in the conflict, provided Union authorities acepted their story.