A similar question has been asked in https://mathoverflow.net/questions/172521/what-morphisms-induce-injective-surjective-maps-on-weil-cohomology, but I have a bit of a variation to the answeres provided there. If $f:A\rightarrow B$ is a surjective morphism of schemes, then $$(*) \quad H^*(B)\rightarrow H^*(A)$$ is injective, where $H$ is a Weil cohomology. Assume that $X,Y$ are smooth projective varieties over a field $k$ with a morphim $X\rightarrow Y$. Are then the morphisms $$H^n(X)\rightarrow \prod_{y\in Y(k)}H^n(X_y)$$ and $$H^n(X\times_k{\bar{k}})\rightarrow \prod_{y\in Y(k)}H^n(X_y\times_k \bar{k}) $$ injective, where $\bar{k}$ is an algebraic closure of $k$? If $$\coprod_{y\in Y(k)}X_y\rightarrow X$$ were surjective, this would follow from $(*)$. However, it can not be true for $2\text{dim} X>n>2\text{dim} X_y$. So is this true for some range of $n$?
PS: The Weil cohomology I care about is actually mostely the $\ell$-adic cohomology, but I'll take what I get.