To achieve promised savings, hospitals turn over the management of their front-line staffing — like patient registration and scheduling — and their back-office collection activities.
Collector Is Faulted for Tough Tactics in Hospitals
I think it does- they hand the job over to someone else to run.
In Br E, we could turn over something to someone, but we have to express the recipient. I expect it means either that they 'outsource' the admin (turn over the admin to outsiders) or that top management lay off middle management and make their front-line workers (nurses, porters and so on) turn their attention to admin (distracting them from from their real jobs).
Wait for the USA to wake up.
[AmE - not a teacher]
"Staffing" is the present participle of to staff, used here as a gerund. "front-line staffing" means the workers (staffing) who interact with the public (front-line).
I am assuming elsewhere in the article it is more clear about who they "turn over" the management to. Pulled out as one sentence, I agree with the others that it seems to need an object of the "turning over." Though the meaning is clear to me, just not the object.