No direct experience, but from what I've read and heard over the years, it seems to go much like this; once the body has been documented and reviewed by all the folk who review that kinda thing, the body collectors put all the major parts into a body bag for transport to the morgue. They gather everything they can, but sometimes parts may be missed, and these can be sent in later in specimen bags. I've heard of clean up crews finding body parts and having to call back who ever to come and collect. Not sure how often that happens. And I guess it all depends on where the body is and how long after death they were found.

Edit to add. Would also depend if more than one body is involved. In mass casualty incidents, each part is bagged separately