Yeah, but reality check: At Geneva station (Cornavin), trains basically run to France (and that platform ain't always open), to Lausanne and beyond (non newsworthy), and GVA. If you follow the crowd at GVA rail, you end up topside at the departure terminal. The first door is France departures, and if a kid with no bags hits security at the right moment of confusion, then she's through. Two gates and a bus. AF and EZ mostly, with a couple of LX departures.
In short: if your kid hops a train at Cornavin, alert the CFF and the staff at gates 60-61.