Does each moment sit on its own?  No past until one is made up.  No future until we form desire.  Karma does seem inherited.  The karma we perceive we have accumulated has an impact on who we think we are in the current moment.  It paints a picture of our past that matches up with our perceptions and it generates a future based on desires.

Karma has a giver and a receiver.  To build up good or bad karma we do good or bad things to other people or ourselves.  When the self is eliminated, so is the karmic cycle.  If the environment (inclusive of ourselves and others) is a projection of the self or the self is part of the environment, then the self is everything (or part of everything) and does not stand alone.  Since it does not stand alone, it cannot be called self.  There is no self.  Without self, there is nothing to accumulate karma.