  We extend the notion of Rokhlin dimension from topological dynamical systems
to $C^*$-correspondences. We show that in the presence of finite Rokhlin
dimension and a mild quasidiagonal-like condition (which, for example, is
automatic for finitely generated projective correspondences), finite nuclear
dimension passes from the scalar algebra to the associated Toeplitz--Pimsner
and (hence) Cuntz--Pimsner algebras. As a consequence we provide new examples
of classifiable $C^*$-algebras: if $A$ is simple, unital, has finite nuclear
dimension and satisfies the UCT, then for every finitely generated projective
$\mathcal{H}$ with finite Rokhlin dimension, the associated Cuntz--Pimsner
algebra $\mathcal{O} (\mathcal{H})$ is classifiable in the sense of Elliott's
Program.
