  In Part I of this work, we proposed a stochastic model describing solute
interactions with stationary and moving grain boundaries (GBs) and applied it
to planar GBs in 1D systems. The model reproduces nonlinear GB dynamics, solute
saturation in the segregation atmosphere, and all basic features of the solute
drag effect. Part II of this work extends the model to 2D GBs represented by
solid-on-solid interfaces. The model predicts a GB roughening transition in
stationary GBs and reversible dynamic roughening in moving GBs. The impacts of
the GB roughening on GB migration mechanisms, GB mobility, and the solute drag
are studied in detail. The threshold effect in GB dynamics is explained by the
dynamic roughening transition, which is amplified in the presence of solute
segregation. The simulation results are compared with the classical models by
Cahn and L\"ucke-St\"uwe and previous computer simulations.
