From hospital to mortuary -- body processing

For a piece of writing I'm working on, I'd like to know how the body is transported from the hospital to mortuary after someone dies in the hospital. Is it transported in a coffin or simply in a body bag. Does the mortuary pick up the body in a hearse? What if the body is transported by air to another location --flown home in other words. How is it transported, in what type of container?

Should I just call a funeral director?