  We look at the time dependent fluctuations of the electrical charge in an
open 1D quantum system represented by a quantum dot experiencing random lateral
motion. In essentially non-adiabatic settings we study both diffusive and
ballistic (Levy) regimes of the barrier motion where the electric current as
well as the net pumped electric charge experience random fluctuations over the
static background. We show that in the large-time limit, $t \to \infty$, the
wavefunction is naturally separated into the Berry-phase component (resulting
from the singular part of the wave amplitude in the co-moving frame) and the
non-adiabatic correction (arising from fast oscillating, slow decaying tails of
the same amplitude). Based on this separation we report two key results:
Firstly, the disorder averaged wave function and current are asymptotically
mainly determined by the same Berry phase contribution that applies in the case
of adiabatic motion. Secondly, after a short transition period the pumped
electric charge exhibits fluctuations that grow much faster than predicted by
the adiabatic theory. We also derive the exact expressions for the average
propagator (in the co-moving basis representation) for the diffusive and
ballistic types of motion considered.
