Recall the notion of an $n$-cateogry $C$ enriched in a symmetric monoidal category. Instead of a *set* of $n$-morphisms $mor(a, b)$ (where $a$ and $b$ are compatible $(n{-}1)$-morphisms), we have an *object* $mor(a, b)$ in some symmetric monoidal category $S$. Composition of $n$-morphisms in $C$ takes the form of an $S$-morphism $mor(a, b)\otimes mor(b, c)\to mor(a, c)$.

I've come across some examples which seem to be well-described by the following generalization of enrichment. We replace the symmetric monoidal category $D$ with an $(n{+}1)$-category $D$. Each $k$-morphism of $C$ is assigned a $k$-morphism of $D$, for $0\le k < n$. (These assignments must of course satisfy various compatibility conditions. I'm only trying to sketch the rough idea here.) If $a$ and $b$ are compatible $(n{-}1)$-morphisms of $C$, $mor(a, b)$ is an $n$-morphism of $D$ (whose domain and range are determined my the assignments for $a$ and $b$). Composition of $n$-morphisms of $C$ takes the form of an $(n{+}1)$-morphism in $D$ from $mor(a, b) \bullet mor(b, c)$ to $mor(a, c)$, where $\bullet$ denotes composition of $n$-morphisms in $D$.

(To recover the more familiar symmetric-monoidal-category-enriched case, let $S$ be a symmetric monoidal category and let $D$ be the $(n{+}1)$-category whose $n$-morphisms are objects of $S$, whose $(n{+}1)$-morphisms are morphisms of $S$, and whose $k$-morphisms are trivial for $k < n$.)

My question is:

Does anyone know of references which discuss versions of enrichment similar to this, or which give natural examples of it?

All I've been able to find so far is Tom Leinster's article Generalized Enrichment for Categories and Multicategories, which discusses a very general version of $n$-categories enriched in $(n{+}1)$-categories. I'm not certain whether what I describe above can been seen as a special case of Leinster's definition.