  In this paper, we study the (in)sensitivity of the Khovanov functor to
four-dimensional linking of surfaces. We prove that if $L$ and $L'$ are split
links, and $C$ is a cobordism between $L$ and $L'$ that is the union of
disjoint (but possibly linked) cobordisms between the components of $L$ and the
components of $L'$, then the map on Khovanov homology induced by $C$ is
completely determined by the maps induced by the individual components of $C$
and does not detect the linking between the components. As a corollary, we
prove that a strongly homotopy-ribbon concordance (i.e., a concordance whose
complement can be built with only 1- and 2-handles) induces an injection on
Khovanov homology, which generalizes a result of the second author and Zemke.
Additionally, we show that a non-split link cannot be ribbon concordant to a
split link.
