I have two questions regarding the homotopy exact sequence associated to étale fundamental groups. Firstly, let me set things up (I'm following Szamuely's book, namely Proposition 5.6.1 and Remark 5.6.3.2).
Let $X$ be a quasi-compact, geometrically integral scheme over a field $k$ [we might as well assume, for relevance to my questions here, that $k$ is finitely generated over $\mathbb{Q}$]. Fix an algebraic closure $\bar{k}$ of $k$ [hence, under our additional assumption, $\bar{k}$ is also a separable closure $k_s$ of $k$] and let $\bar{X} = X_{\bar{k}}$ denote the base change of $X$ by $\text{Spec}(\bar{k})\to\text{Spec}(k)$. Let $\bar{x}:\text{Spec}(\bar{k})\to\bar{X}$ denote a geometric point of $\bar{X}$ and let $$x : \text{Spec}(\bar{k})\xrightarrow{\bar{x}}\bar{X}\xrightarrow{p_X} X$$ be the geometric point of $X$ obtained from $\bar{x}$ via the natural morphism. The homotopy exact sequence of étale fundamental groups (with respect to these basepoints) is an exact sequence of étale fundamental groups $$1\to\pi_1 (\bar{X},\bar{x})\to\pi_1(X,x)\to \text{Gal}(\bar{k}/k)\to 1$$ where the homomorphisms between fundamental groups are induced by the natural maps $\bar{X}\to X\to\text{Spec}(k)$.
1) My first question concerns the proof of exactness of this sequence (specifically, exactness in the middle), although it is really just a geometrical question and not directly related to fundamental groups. In the proof of Proposition 5.6.1 Szamuely obtains two finite étale Galois covers $Y\to X$ and $X_L\to X$ (where $L/k$ is a finite Galois extension) having the same function field. Thus $X_L$ and $Y$ are isomorphic over the generic point of $X$, so there is a dense open $U\subseteq X$ so that $X_L \times_X U\cong Y\times_X U$. Szamuely then says that local freeness of the covers $Y\to X$, $X_L\to X$ (being étale) implies that $Y\cong X_L$ i.e. that this isomorphism extends over the whole of $X$. I don't understand how local freeness is used to obtain this isomorphism. Could somebody explain this?
2) My second question concerns the role of the basepoint in Remark 5.6.3.2, on the Section Conjecture. Recall any $k$-point of $X$ can be described as a section $$y: \text{Spec}(k)\to X$$ of the structural morphism $t:X\to\text{Spec}(k)$. A choice of algebraic closure then composes to give a geometric point $$\bar{y}:\text{Spec}(\bar{k})\to\text{Spec}(k)\to X.$$ Let $\sigma_y : \text{Gal}(\bar{k}/k) \to \pi_1 (X,\bar{y})$ denote the induced homomorphism on fundamental groups. Then $\sigma_y$ isn't quite a splitting of the homotopy sequence because $\pi_1 (X,\bar{y})$ and $\pi_1 (X,x)$ are, technically, different groups. However, they are isomorphic via $$\lambda : \pi_1 (X,\bar{y})\to\pi_1 (X,x)$$ which is unique up to conjugation by an element of $\pi_1 (X,x)$. Szamuely then claims that $\lambda \circ\sigma_y$ is a section of the homomorphism $t_*:\pi_1 (X,x)\to \text{Gal}(\bar{k}/k)$.
I don't understand why this is the case and why $t_* \circ\lambda\circ\sigma_y$ is not an arbitrary automorphism of $\text{Gal}(\bar{k}/k)$. The question boils down to whether the isomorphism $\lambda$ induced by an étale path (an isomorphism of the fibre functors $F_{\bar{y}}\xrightarrow{\sim}F_x$) is compatible with the structural morphism $t:X\to\text{Spec}(k)$. i.e. let $t'_*:\pi_1 (X,\bar{y})\to \text{Gal}(\bar{k}/k)$ denote the induced homomorphism on fundamental groups with basepoint $\bar{y}$. Then is $t'_* = t_* \circ\lambda$?