## limit of étale cohomology of representable sheaf

Let $\mathcal{F}$ be a representable sheaf on a smooth irreducible scheme $X$ and $Y \hookrightarrow X$ be an integral reduced closed subscheme with generic point $\eta_Y$. How does $H^n_Y(X,\mathcal{F})$ compare to $H^n_{\eta_Y}( \mathcal{O}_{X,\eta_Y},\mathcal{F})$ (étale cohomology), i.e. is there an injective map in some direction? I want to apply excision, but this forces me to leave $Y$ fixed; and I also don't see how to apply Milne, Étale cohomology, Lemma III.1.16 for the same reason.

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