Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid completion of the category of points) of $Sh (X_{ét})$, $Pt (X_{ét})$, is described by geometric points as objects and paths of étale specialisations and generalisations between them (SGA4). For the corresponding (pro)-étale $\infty$-topoi, $Sh_{\infty} (X_{(pro)-ét})$, the $\infty$-groupoid of points is $1$-truncated since $Sh_{\infty} (X_{(pro)-ét})$ is $1$-localic and, hence, it doesn't maker if one works with the respective $\infty$-topoi. 

  On the other side, $Pt (X_{pro-ét})$ has geometric points as points, however such collection is not even conservative (https://stacks.math.columbia.edu/tag/0991).

  Now, there's also the shape of $Sh_{\infty} (X_{ét})$, $\Pi_{\infty} (X_{ét})$, and its profinite version, $\widehat{\Pi}_{\infty} (X_{ét})$, which coincides with the étale homotopy type of $X$. When $X$ is qcqs, it's also known that $\Pi_{\infty} (X_{ét})\cong \Pi_{\infty} (X_{pro-ét})$ (6.1.6 in https://arxiv.org/pdf/1807.03281.pdf). 

  Let $\pi^{BS}_1 (X, \overline{x})$ be defined as the automorphisms of a fiber functor from étale coverings satisfying the valuative criterion for properness (https://arxiv.org/pdf/1309.1198.pdf). In $Sh (X_{pro-ét})$, there are several points that are not geometric (as mentioned above) and, in fact, they are not even a conservative family. On the other side, $\pi_1 ({\Pi}_{\infty} (X_{ét}), \overline{x})$, when $X$ is connected locally Noetherian, has dense image in $\pi^{BS}_1 (X)$ (Rem 7.4.12 https://arxiv.org/pdf/1309.1198.pdf).

  1) What's the relation between $\pi^{BS}_1 (X, \overline{x})$ and $Pt (X_{pro-ét})$? Is it just the connected component of $\overline{x}$?

  2) Is every  element of $\pi^{BS}_1 (X, \overline{x})$ given by a path of specialisations and generalisations?

  3) What are the points and paths in $Pt (X_{pro-ét})$? I think the points are given by connected affine $w$-contractible objects, i.e., connected $w$-strict local rings, i.e., a local ring having a section for every affine fpqc covering.


  Now, let's purposefully complicate everything a little more. For $X$ qcqs, one can define a profinite stratified space or, equivalently (Hochster's duality), a spectral stratified $\infty$-topoi ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{(pro)-ét})}$ (https://arxiv.org/pdf/1807.03281.pdf). By taking the groupoid completion, one gets back $\Pi_{\infty} (X_{(pro)-et})$ (recall that, as mentioned above, changing from étale to pro-stale gives an equivalent shape) and, by taking the materialisation (I guess it means just taking the real limit in spaces of a profinite stratified space, ), one gets a $X_{Zar}$-stratified version of $Pt (X_{(pro)-et})$ before the groupoid completion. 

 4) Do the materialisations coincide at first or after inverting all the maps?

 5) What's really the difference between ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{ét})}$ and ${\widehat{\Pi}_{(\infty, 1)}^{X_{Zar}} (X_{pro-ét})}$?

  Maybe I've screwed up something in my assertions. If so, please, comment below.

  Thanks in advance.