Once again sorry for the formatting, I'm using a phone.

Fix an étale cover of $Y\times S$, where $S$ is connected. We pull-back along inclusions of points into $S$ to get a family of étale covers of $Y$. Are the Galois groups of these covers isomorphic? This seems unlikely to be true but I'd love if it were. A reference would be ideal.

geometricpoint $\text{Spec} \ \kappa \to S$, you compute the Galois group of the finite etale cover of $\kappa$-schemes $X_{\kappa} \to Y\times_{\text{Spec}\ k} \text{Spec}\ \kappa.$ That would eliminate the issue of the previous commenter. If that is what you mean, please confer SGA 1, Exp. X, Cor. 2.4, Thm. 3.8, Cor. 3.9 and Section 2 of Exp. XIII. $\endgroup$