Let $S$ be a noetherian scheme and let $X$ be a "nice" algebraic stack over $S$. For instance, let's say $X$ is a finitely presented algebraic stack over $S$, or that $X$ is a finite type separated DM-stack over $S$.

Let $P$ be an absolute or relative property. More precisely, let $P$ be "flat over $S$", "smooth over $S$", "etale over $S$", "Deligne-Mumford over $S$", "regular algebraic stack", "normal algebraic stack", "reduced algebraic stack", "separated over $S$", "proper over $S$", ...

Assume $X$ has $P$. Does $I_X$ have $P$?

I know that these are a lot of questions in one. I am only looking for hints on some of the difficult ones to verify, so that I (hopefully) get a feeling how to verify others myself.

Example. If $X$ is DM over $S$, then $I_X\to X$ is DM.

Example. If $X$ is proper over $S$, then $I_X\to X$ is proper and thus $I_X$ is proper over $S$.

Example. If $X$ is separated over $S$, then $I_X\to X$ is proper and thus $I_X$ is separated over $S$.

If $X$ is smooth (flat, etale, unramified) over $S$, is $I_X$ smooth (flat, etale, unramified) over $S$?

Edit: What if $S$ is the spectrum of a field?