The absolute cohomological purity theorem in étale cohomology is as follows.

Let $X$ be a regular scheme over $\mathbb{Z}[1/n]$, and $i \colon Z \to X$ the inclusion of a regular closed subscheme everywhere of codimension $c$. Let $\Lambda$ be the constant sheaf $\mathbb{Z}/n\mathbb{Z}$ on $X$. Then $R^{2c} i^! \Lambda \cong \Lambda(-c)$, and $R^{p} i^! \Lambda$ is trivial for $p \neq 2c$.

Here $i^!$ is the functor taking a sheaf $\mathcal{F}$ on $X$ to $\mathrm{ker}(\mathcal{F} \to j_* j^* \mathcal{F})$, considered as a sheaf on $Z$, where $j$ is the inclusion of $X \setminus Z$. There are several other ways to state the result: chapter 16 of Milne's online Lectures on Etale Cohomology explains nicely how to go between them. As far as I understand, the theorem has been proved by Gabber in at least two ways.

My question is this:

What other sheaves $\mathcal{F}$ may replace $\Lambda$ in the purity theorem?

The problem seems to be in defining the morphism $R^{2c} i^! \mathcal{F} \to i^* \mathcal{F}(-c)$ in the first place: once that's done, proving that it's an isomorphism is purely local and ought to follow from the case of $\Lambda$, at least for $\mathcal{F}$ something nice like a flat sheaf of $\Lambda$-modules. Maybe one can define the morphism just by tensoring the original one with $\mathcal{F}$, but I'm not sufficiently happy with the formal properties of étale cohomology to understand how $R^p i^!$ works with tensor products.