The belief that the elaborate coffin gets burned with the body is incorrect. With few exceptions, the coffin does not get burned. Before incineration, the body will be transferred to a thin wooden or cardboard box, in some cases, even just laid on a metal tray and covered with a veil. Also, the cremation process can vary… religious protocols or crematorium practices may or may not remove the clothes of the deceased. After cremation, it is normal that some large bone fragments remain; these will be mechanically ground to dust. The ashes placed in the urn are pretty much all the ashes of the body.
It is to be noted that above the 800ºC high temperatures in the cremation chamber, most of the soft tissues and organic matter will evaporate away through the chimney, leaving behind primarily bone fragments and some minerals.