DFW/CLT off road adventure

That still does not answer the question, do they have a responsibility at the very to least notify someone that they may have pushed debris up on an active runway?
 
do they have a responsibility

A responsibility such as a regulation? Report it to the tower? To the best of my knowledge not in a case of just going off the runway but getting back on it and taxiing to the ramp That could be faulty memory though, so don't take it as fact. I suspect that most airlines have a policy requiring a report to at least dispatch or maintenance.

Would something normally be said to the tower? Yes, although there have been cases where the crew kept quiet hoping no one noticed. We had a case at US back in the early 90's - a DC9 crew landing on 32 ran off the end at low speed, got on the taxiway that becomes the parallel to 27R, taxied to the ramp, deplaned the passengers, shut the airplane down, and went home without saying anything to anyone or putting anything in the maintenance log. Needless to say, maintenance found the gear caked with mud (no damage done) when they did the overnight inspection, started investigating, and the airport found the tracks in the mud between the end of 32 and the taxiway. The crew ended up in hot water with the company over that one. If they'd just told the tower, wrote it up, and called dispatch they'd probably only gotten a "Don't let it happen again" lecture.
Jim