The Greek word translated as appoint is a compound verb that literally means to stretch out a hand.
The idea is the approval of a candidate through the show of hands in the context of a meeting.1 This implies that the elders were selected from among the church members, and that there was a vote by church members, similar to appointments elsewhere in Acts (see Acts 1:23–26; Acts 6:3–6).2 Paul and Barnabas did not decide who the elders will be (a top-down approach), but they organized and arranged for their election, giving instructions regarding their task and who can serve in the office (see 1 Timothy 3:1–16).
23 Ils désignèrent des anciens dans chaque Eglise, et, après avoir prié et jeûné, ils les recommandèrent au Seigneur, en qui ils avaient cru.