In what follows, all schemes are qcqs. Also, let $\operatorname{\acute{E}t}(X)$ denote the petit étale topos of a scheme $X$.  

Let $Y\to X$ be an $X$-scheme.  Say that $Y$ is a **special** $X$-scheme if for any $X$-scheme $Z\to X$, the universal geometric morphism

$$\operatorname{\acute{E}t}(Y\times_X Z)\to \operatorname{\acute{E}t}(Y)\times_{\operatorname{\acute{E}t}(X)} \operatorname{\acute{E}t}(Z)$$

is an equivalence of toposes.  

Is there a known characterization of the special $X$-schemes for a scheme $X$?

I know of two classes of $X$-schemes that satisfy this property:

* If $Y\to X$ is a closed immersion, then $Y$ is a special $X$-scheme
* If $Y\to X$ is pro-étale and qcqs, then $Y$ is a special $X$-scheme (for qcqs étale morphisms, this is obvious, as it follows from slicing of sites, and to get pro-étale morphisms, apply some version of Noetherian approximation)

Are there any other kinds of maps that satisfy this property? What about in the case where $X$ is Noetherian?