We are responsible for the present. This is clear. How we act hurts or helps our parents, siblings, neighbors, and fellow humans.
We are responsible for the future. This is clear. How we act “echoes in eternity,” and we shape the future world for future people in what we do (and fail to do now).
Something less imagined is our obligations to the (former) lives of the dead.
Admittedly this is where things get a little less literal and a little more metaphorical, but there is a sense in which we are responsible for the lives that predated us. For to the extent we maintain and improve their work, we keep them alive and present with us.
On my family’s farm and in our business, when I use a tool my grandfather used or use a structure he raised, I give some tangible partnership to his memory. Just as a statue or a book gives a man “immortality”, these forms of more mundane remembrance keep him alive, at least in my memory. In fact, in continuing the work of a former human, we do more to extend their footprint than even the finest statue could do.
Assuming we all want some form of immortality, it seems just and decent to reward it to our good forebears. You could argue this is a sort of duty.