These captured flags were initial turned over to the Commander of the forces of Major General John C. Breckenridge.

I would assume by the below that they were forwarded to the Confederate Adjutant General in Richmond.

Then, they were most likely lost during the evacuation of Richmond along with army and government archives.

George Martin

Assistant Adjutant-General.
ADDENDA.
HDQRS. DEPT. OF WESTERN VIRGINIA AND EAST TENN,
Wytheville, October 19, 1864.
Brig. Gen. J. C. VAUGHN,
Commanding, &c.:
GENERAL: The 2 flags captured(*) from the Eighth and Ninth Federal Tennessee Cavalry have been received, together with the communication accompanying them. The major-general commanding desires you to convey to your command his thanks for the gallantry displayed in the action in which they were captured, and his special acknowledgments of the signal valor and daring of Privates Samuel H. Selvidge, Company C, Third Tennessee Regiment, and Charles Dent, Company B, Sixteenth Tennessee Battalion, in securing the enemy's colors. The flags will be forwarded with the statements sent by you appended, and they will be recommended for promotion.
I am, general, very respectfully, your obedient servant,
J. STODDARD JOHNSTON,
Assistant Adjutant-General. (OR 39/1, 565/6)