From studying the rearming of the Trans Mississippi Department after Van Dorn's robbery of everything he could to take from Arkansas, the arming of troops in the west was not a priority of the Richmond Government. Hence I would figure that the Springfields and Lorenz rifle stayed close to Richmond until after the Seven Days Battles and 2nd Manassas from which the Army of Northern Virginia was very well equipped.
In August of 1862 I did find where one of the commanders of either the 24th or 25th Texas cavalry dismounted was able to travel to Richmond and came back to White Sulphur Springs with 100 1862 Richmond rifles. But that was and oddball incident and didn't follow the pattern of hand-me-down weapons travel from East to West from Richmond. That is what the CSS Fairplay was carrying when it was captured above Vicksburg by Union gunboats was 5,000 arms bound for the Trans Mississippi from the castoff weapons that had been returned to armouries after the Seven Days and 2nd Manassas Battles.