I have a question. Maybe it be trivial, but I'm cannot conclude nothing yet.
Suppose that we have a free profinite product $$G = G_1 \amalg \cdots \amalg G_n.$$ By definition in Ribes-Zalesskii $G$ is a profinite group. If all $G_i$ becomes to another category, is also $G$? I'm interested in the case when each $G_i$ is a pro-$p$ group (same $p$ for all $G_i$). Will be $G$ a pro-$p$ group?
No. For instance, $S_3$ is generated by two transpositions, so the profinite free product of two cyclic groups of order 2 is not a pro-2-group since it has an epimorphism to $S_3$.