I have had excellent experience with Delta in such situations - with tickets being issued/changed due to these circumstances. I have been a DL Platinum in both cases (deaths of my parents) and a call to the DL resulted in helpful and courteous service. In one case a ticket was issued at a very reasonable price (I was purchasing a ticket for next day travel from Little Rock to Rome) with a no-cost for change of return dates added to the notes on my reservation. In another case my itinerary was changed at no cost to me. In both cases I was asked to provide the name of the deceased, location of death and the hospital where it occurred (this latter point was waived when I explained that both cases happened outside of the US).
These two instances are one of the reasons that when provided the opportunity to chose, if within reason on fare price, I try to fly with Delta.