  This work establishes a condition that determines when strong generation in
the bounded derived category of a Noetherian $J\textrm{-}2$ scheme is preserved
by the derived pushforward of a proper morphism. Consequently, we can produce
upper bounds on the Rouquier dimension of the bounded derived category, and
applications concerning affine varieties are studied. In the process, a
necessary and sufficient constraint is observed for when a tensor-exact functor
between rigidly compactly generated tensor triangulated categories preserves
strong $\oplus$-generators.
