A closed natural transformation is the appropriate sort of natural transformation relating closed functors between closed categories.
Let $F,G\colon C\to D$ be closed functors. A closed natural transformation $\alpha\colon F\to G$ is simply an ordinary natural transformation such that:
The following diagram commutes:
The following diagram commutes for any $X,Y$:
Just as closed functors between closed monoidal categories are in bijective correspondence with (lax) monoidal functors, closed natural transformations between such are in bijective corrrespondence with monoidal transformations.
The same is true for transformations between closed unital multicategories, regarded as closed categories.