Here is an example which I like to have a name for.

Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary.

Define $E(k,P)$ to be the space of smooth (codimension zero) embeddings $$ \coprod_{k} D^p \to P \, , $$ that is the space of embeddings of $k$ disjoint $p$-disks in $P$, where the image of each such embedding lies in the interior.

In particular, when $P = D^p$, the spaces $\{E(k,D^p)\}_{k\ge 0}$ form an operad.

There is an evident "action" map $$ E(\ell,P) \times (E(k_1,D^p) \times \cdots \times E(k_\ell,D^p)) \to E(k_1 + \cdots +k_\ell,P) $$ given by insertion.

Question: What is this action an example of? (Does it have a name?)