Relief duty was the duty of replacing a prior group in their assignment.
In 2268, Captain James T. Kirk ordered a security detachment from USS Enterprise to prepare for relief duty on Triacus, replacing the team he'd left to guard the planet previously. Unfortunately, when the relief team beamed away, the Enterprise was no longer orbiting Triacus, so they were deposited into empty space and died. The team they were to replace was not able to be retrieved. (TOS: "And the Children Shall Lead")