Ella Lonn in her epic work on the subject of Desertion, covers this in Chapter V, "Deserter-Hunting."

"The method pursued for hunting down deserters ranged from spy-work to bloodhounds."

".... dogs were kept at all the prisons, picket posts, and guard stations to chase escaped prisoners, Union or Confederate; especially were they to be found at the ferries and fords."

The killing of dogs on the hunt demanded death under military law. [Lonn]

Ella Lonn, Desertion During the Civil War, Univ. of Nebraska Press, 1998