In this paper, we study the (in)sensitivity of the Khovanov functor to 4-dimensional linking of surfaces. We prove that if (Formula presented.) and (Formula presented.) are split links, and (Formula presented.) is a cobordism between (Formula presented.) and (Formula presented.) that is the union of disjoint (but possibly linked) cobordisms between the components of (Formula presented.) and the components of (Formula presented.), then the map on Khovanov homology induced by (Formula presented.) is completely determined by the maps induced by the individual components of (Formula presented.) and does not detect the linking between the components. As a corollary, we prove that a strongly homotopy–ribbon concordance (that is, 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.