  We present a class of tractable non-equilibrium dynamical quantum systems
which includes combinations of injection, detection and extraction of particles
interspersed by unitary evolution. We show how such operations generate a
hierarchy of equations tying lower correlation functions with higher order
ones. The hierarchy closes for particular choices of measurements and leads to
a rich class of evolutions whose long time behavior can be simulated
efficiently. In particular, we use the method to describe the dynamics of
current generation through a generalized quantum exclusion process, and exhibit
an explicit formula for the long time energy distribution in the limit of weak
driving.
