Comrades might sent bodies home, this would however usually occur if they were serving at a camp, but not from battlefields. Families would occasionnly retrieve bodies for return home.
Here's an interesting piece:
Richmond Daily Dispatch.
Thursday morning...July, 31, 1862.
Disinterment of dead bodies.
We daily observe at the railway stations boxes containing the bodies of deceased soldiers, which have been disinterred by their friends, under the belief that they can be sent off without delay, either by mail train or express. This, however, is an error. Freight trains only carry them, and the detention frequently causes the bodies to become offensive, when their immediate burial by the way side is a matter of necessity. It would be better to postpone disinterment until cold weather, when it can be accomplished with less trouble and more certainty of getting the remains of the departed to their destination. Metallic coffins are difficult to obtain, and wooden ones can only be procured by the payment of a large sum. In these the dead bodies are packed with sawdust, and in warm weather their transportation to a distant point is uncertain, if not absolutely impossible.