Relations and interactions between the theorems of Ash, Herwig--Lascar and Ribes--Zalesskii are discussed and it is shown that these three theorems are equivalent in the sense that each of them can be derived from each other one. Some strengthenings of these theorems are obtained with the use of groups provided by a construction of the third author. Evidence is given that these strengthenings are substantially stronger than the classical results. Yet, it turns out that both kinds of results can be interpreted as different instances of the same common scheme, namely as \emph{finite approximation of free groups}.