Delta did this for me, I called in a panic because the flight was supposed to depart in 30 minutes and I was sitting in my office when I got then news. They just booked me on the flight and the people (even TSA) was aware of my situation. I picked up my ticket was escorted to the front of the TSA line, and the plane was waiting for me. Someone may have even given up a seat. From sitting in my office in Missoula MT to standing by my dads hospital bed I Memphis TN was about six hours. I was only 30 minutes behind my mother. Delta didn't even charge me for the flight, I was in such a rush and such a panic that they didn't bother with technicalities.
With that, I got to spend one last night with my dad. The next flight out, he would have been gone. United on the other hand saw my plight as an opportunity to capitalize AND DID. This was on July 31st 2012.