In my state the coroner never goes to the crime scene. LE does and the CSI folks and then one or 2 representatives from the M.E. office will come to collect the remains and transport them back to the M.E. office for autopsy, which is usually done within 24 to 48 hrs. No medical examination is done at a scene--that all happens in the morgue's examining room.
There was a case locally in which the murder happened in a nearby city but the deceased was then dismembered and transported to TX and her parts were scattered in a pond. Divers in TX found and collected the remains (at least what pieces could be recovered), and transported those to the University of TX in a bag, which was contained inside a box where the initial autopsy took place.
Those same remains were later transported back to NC for a forensic anthropologist working for the state to do an examination of the bones to determine the type of cut marks and tools that would match up. Those bones (skeletal pieces) were transported in (you guessed it), a box, which was delivered by secure courier to the State University lab.
During the OJ murder case (this is now CA), you can see video of the remains of Ron & Nicole being collected (in white sheets, then put in body bags) then rolled on gurneys to the truck for transport. The M.E. never went to the scene itself. The remains go to the morgue and are examined there.