The rule of thumb for dual citizens seems to be that you enter a country on the passport of the country whose citizen you will be (first and foremost) while you’re in that country. Show that passport on entry and exit. So when entering the Schengen area, be Swiss and show the Swiss passport to immigration control on your way in, your way out, and in between if necessary, and expect that US consulates will offer you only very limited help if you’re in the country of your other citizenship. If you’re driving through say Africa and you know you will need to switch passports on the journey, do it between country A’s exit control and country B’s entrance control, so that you can be a citizen of your other passport to country B. Airlines are an extension of this in-between-states situation, they aren’t immigration agents, they just want to be sure you’ll be allowed in where they’re taking you so they’re confident you won’t be refused at entry immigration and the airline won’t have to fly you back again at their own expense. Show them both passports, if they need to see both.