  We study random compositions of transformations having certain uniform
fiberwise properties and prove bounds which in combination with other results
yield a quenched central limit theorem equipped with a convergence rate, also
in the multivariate case, assuming fiberwise centering. For the most part we
work with non-stationary randomness and non-invariant, non-product measures.
Independently, we believe our work sheds light on the mechanisms that make
quenched central limit theorems work, by dissecting the problem into three
separate parts.
