The detail that Anthony is providing you is obtained by reading the Compiled Service Records for the Soldiers. In the late 1890's clerks at the War Department were detailed to create service records on all former Soldiers. These records, known as CSRs were developed from the Muster Rolls for the various units. The actual muster rolls really aren't available. The CSRs were later placed on Microfiche. Those Microfiche are available an many public libraries. More recently several commercial services have made those records available on line. This message board was supported, in part, by fees paid for copies of service records. The CSR's contain various types of cards. Generically, Officers have more complete CSRs than Enlisted men.

The Edward G. Gerdes Civil War Home Page, has a lot of information, including a very good explination of the term deserted:

"FEW WORDS about the term "deserted". Many of these soldiers have the word "deserted" after their names.. We would like you to know that this is from the Microfilm from the National Archives. These soldiers may have been separated from their Units in the heat of battle, joined up with other Units and continued to fight this war.. They may have had families back home who were starving and no one to plant their crops or gardens... They may have gone home to visit awhile with loved ones and then return, only to find that there were so many miles between them and their original Unit that rejoining them was an impossibility, and so they joined with others to continue to fight in the war. " http://www.couchgenweb.com/civilwar/

In my own family, John and Jacob Gordon, who fought at Prairie Grove and Helena as part of Hart's 30th Arkansas, left (deserted) from the unit immediately following the Battle of Helena. They were as close to home as they had been since enlistment. It appears that they went home, too care of crops and family, and then joined new Cavalry units and participated in Price's Raid, along with your ancestors.

Another good place to start is the National Park Service Soldier and Sailor:
http://www.nps.gov/civilwar/search-soldiers.htm

The Gerdes site and Wikipedia have histories of all of these unit.