The data you have on my great-grandfather, Elihu C. Graham, matches what I have but I am very surprised to learn his brother M.C. was listed as deserting. My data does show him enlisting with his brothers in March, 1862. I do not have any muster roll indicating his being detailed as a teamster carrying ordnance. Does your muster roll indicate the location and date of this duty? More importantly, in whose unit and where was it located at the time of his desertion? I do have only one muster roll and it's apparently the same one you are referencing about his unit surrendering on 11 May 1865.

I really question the "desertion." First, that last muster & descriptive roll states it's "of prisoners of war belonging to the Army of the Northern Sub-District of Arkansas." Could it be that he was captured rather than deserting on June 26, 1863? Do you know if his unit was engaged in battle on that date? If he had deserted in 1863, then re-joined his unit 2 years later for the surrender, wouldn't his commander have immediately taken him as a prisoner for the Confederacy? Hard to make sense of it.
Further, M.C. applied for a Confederate pension in 1899 while living in Texas. Of course, he had multiple fellow soldiers to support his application swearing that he served throughout the war, never deserting. On submission, the comptroller verified his service record without any reference to desertion. His application was "denied" but it was because he had sold some farm land to a son prior to asking for a pension so it was seen as trying to get rid of assets so as to be eligible for pension.

Could you please explain further about John C. Graham as I have only that he enlisted with his brothers in March 1862 as 2 Lt. I don't have any document indicating a promotion from 3rd Lt. or about a unit reorganization at Corinth. Do you have a muster roll indicating he was not "retained"? What exactly does that mean as I have never heard of that? Could he just have decided not to re-enlist and returned home?