  We present a detailed analysis of the dynamics of Bloch oscillations of
Bose-Einstein condensates in disordered lattice potentials. Due to the disorder
and the interparticle interactions these oscillations undergo a dephasing,
reflected in a damping of the center of mass oscillations, which should be
observable under realistic experimental conditions. The interplay between
interactions and disorder is far from trivial, ranging from an
interaction-enhanced damping due to modulational instability for strong
interactions, to an interaction-reduced damping due to a dynamical screening of
the disorder potential.
