  This is the first part of a project aimed at formalizing Rozansky-Witten
models in the functorial field theory framework. Motivated by work of
Calaque-Haugseng-Scheimbauer, we construct a family of symmetric monoidal
$(\infty,3)$-categories parametrized by an $\infty$-category with finite limits
and a functor into symmetric monoidal $\infty$-categories, such that the
functor admits pushforwards. This $(\infty,3)$-category contains
correspondences in the base $\infty$-category equipped with local systems,
which compose via a push-pull formula. We apply this general construction to
provide an approximation to the $3$-category of Rozansky-Witten models whose
existence was conjectured by Kapustin-Rozansky-Saulina; this approximation
behaves like a "commutative" version of the conjectured $3$-category and is
related to work of Stefanich on higher quasicoherent sheaves.
