gerard wrote:You'll need to clear immigration in the USA even if you are only transiting through it, which means you'd need to get ESTAs using your EU passports, assuming you're on the list. It's easy enough and you do it all online, but there is a $14 per person cost. They last for 2 years and you can't even check-in if you haven't got them.
Other than that, you'd need a visa. The carriers themselves shouldn't care which passports you're using at any stage provided they have the valid ESTA/visa.
They don't care, and they are pretty used to handling people with double nationalities, as are American immigrations officials. I know my parents were asked to produce their Peruvian Passports entering Miami (they were entering as EU nationals with ESTAs and everything), they showed the Peruvian Passports, the guy looked at them, and gave them back. Easy peasy, I guess he was just curious, perhaps because both my parents have had American visas on their Peruvian Passports?