Flags captured by U.S. troops were supposed to be forwarded to the War Department. Many ended up in the custody of the State whose regiments captured them.
I don't believe the flag of the 5th/13th Arkansas Infantry was captured at Jonesboro, but it was captured by the 88th Illinois Infantry at Franklin later that year. The 5th/13th Arkansas flag, one of five captured by the 88th Illinois at Franklin, was not forwarded to the War Department. It was last seen at Nashville when the 88th Illinois displayed the captured flags to General Thomas. The division commander stated, "They were afterward sent home by those who captured them. Since then they have been ordered to be returned, and will be forwarded to department headquarters as soon as they arrive." However, none were ever returned.
The man credited with capturing the flag of the 5th/13th Arkansas Infantry was Corporal Benjamin Newman, Co. G, 88th Illinois Volunteers. The flag was erroneously reported as the flag of the "Fifth and Sixth Arkansas". The 5th Arkansas was consolidated with the 13th Arkansas at Franklin, and the 6th Arkansas was consolidated with the 7th Arkansas. The 6th/7th Arkansas flag was captured at Jonesboro.