  We study quantitative equidistribution in law of affine random walks on
nilmanifolds, motivated by a result of Bourgain, Furman, Mozes and the third
named author on the torus. Under certain assumptions, we show that a failure to
having fast equidistribution is due to a failure on a factor nilmanifold.
Combined with equidistribution results on the torus, this leads to an
equidistribution statement on some nilmanifolds such as Heisenberg
nilmanifolds. In an appendix we strengthen results of de Saxce and the first
named author regarding random walks on the torus by eliminating an assumption
on Zariski connectedness of the acting group.
