Here is a classification: **Natural operations $\tau : P(-)^n \to P(-)$ correspond to maps $M : P([n]) \to P([n])$ that are *deflationary*, i.e. $M a \subseteq a$ for each $a$.**

A deflationary map $M$ on $P([n])$ induces an operation $\varphi_M : P(X)^n \to P(X)$ by taking $\varphi_M(U_1,\ldots,U_n) = \bigcup_{i \in M(\{ j | U_j \neq \emptyset \})} U_i$. Conversely, a natural operation $\tau$ induces a map $D_\tau : P([n]) \to P([n])$ by sending $a \in P([n])$ to $\tau(a \cap \{1\}, a \cap \{2\}, \ldots)$. This is deflationary by naturality: for each $a \in P([n])$, the tuple $(a \cap \{1\},\ldots)$ is in the image of $P(a)^n \hookrightarrow P([n])^n$, and so its image under $\tau$ is in the image of $P(a) \hookrightarrow P([n])$.

Now certainly for any deflationary map $M$, $D_{\varphi_M} = M$ (by direct calculation).  Conversely, for any natural operation $\tau : P(-)^n \to P(-)$, naturality implies that $\varphi_{D_\tau} = \tau$ as follows: Given $U_1,\ldots,U_n \subseteq X$, write $a = \{ i \in [n] \mid U_i \neq \emptyset \}$, and $\vec a = (a \cap \{1\}, \ldots, a \cap \{n\}) \in P([n])^n$.  We need to show that $\tau(U_1,\ldots,U_n) = \bigcup_{i \in \tau(\vec a)} U_i$.

Take the disjoint sum $U = \coprod_i U_i = \{ (i,x) \mid i \in [n],\, x \in U_i\}$, with natural maps $U \to [n]$ and $U \to X$ given by first and second projections. Write $\vec U$ for $(\{1\} \times U_1,\ldots, \{n\} \times U_n) \in P(U)^n$.  Now the image of $\vec U$ under first projection $P(U)^n \to P([n])^n$ is exactly the set $\vec a$ from above; so the image of $\tau(\vec U)$ under $P(U) \to P([n])$ must be exactly $\tau(\vec a)$.  So $\tau(\vec U) \in P(U)$ must have non-intersection with the summand $\{i\} \times U_i$ precisely when $i \in \tau(\vec a)$.  Moreover, in that case, it must contain the whole summand, by naturality under all permutations of $U_i$.  So $\tau(\vec U)$ consists precisely of the union of all summands $\{i\} \times U_i$ for $i \in \tau(\vec a)$.  So now by naturality under the map $U \to X$, we get that $\tau(U_1,\ldots,U_n)$ is the union of these $U_i$, as required.

This proves the claimed correspondence between natural $n$-ary operations and deflationary maps on $P([n])$.  For counting, a little calculation shows that there are $2^{n2^{n-1}}$ such maps.

This correspondence also restricts to a monotone version: an operation will be monotone precisely when its corresponding deflationary map is.