The men captured at the South Side Railroad were transported to City Point, Va. which was located NE of Petersburg and connected by rail. Believe they were captured by the 68th Pennslyvania which was ordered from Petersburg to City Point April 2. Capt. L. Jackson of that regiment is listed as releasing Private Tulloch for Hart's Island.
From there they were sent to Hart's Island arriving April 7. I would assume they would have to have been sent by ship as there was no other RR from/to City Point.