The pictures below were made in a Moscow morgue that accepts corpses of old people who died a natural death. The morgue is related to a couple of districts only, that's why it accepts zero-several bodies a day.
So, let us follow the true last path of a person. It all starts with death that can occur at home or in a hospital. Condition of the corpse that is taken to a mortuary will be determined by the fact how fast it was discovered by relatives or other people.
Then an ambulance and the police are called. The doctor states the fact of death and the body is taken to the mortuary. If a person dies in a hospital, no police are usually invited.