  The deposition dynamics of particles (or the growth of a rigid crystal) on a
disordered substrate at a finite deposition rate is explored. We begin with an
equation of motion which includes, in addition to the disorder, the periodic
potential due to the discrete size of the particles (or to the lattice
structure of the crystal) as well as the term introduced by Kardar, Parisi, and
Zhang (KPZ) to account for the lateral growth at a finite growth rate. A
generating functional for the correlation and response functions of this
process is derived using the approach of Martin, Sigga, and Rose. A consistent
renormalized perturbation expansion to first order in the non-Gaussian
couplings requires the calculation of diagrams up to three loops. To this order
we show, for the first time for this class of models which violates the the
fluctuation-dissipation theorem, that the theory is renormalizable. We find
that the effects of the periodic potential and the disorder decay on very large
scales and asymptotically the KPZ term dominates the behavior. However, strong
non-trivial crossover effects are found for large intermediate scales.
