"the European Airlines (Alitalia, BA, Iberia) leave from the international terminal."
Important correction - at JFK there are multiple "international terminals." In fact, at at this time, all the terminals except Terminal 2 can process international arrivals. So, you always have to check which terminal your airline is using.
If you go to the JFK website airline list http://www.panynj.gov/airports/jfk-airlines.html, you'll see that at this time, Alitalia uses Terminal 1, British Air uses Terminal 7, and Iberia uses Terminal 7. Which airline uses which terminal does change around, although the "biggies" like British Air or Air France, with multiple flights a day, are more predictable.
When airlines are allied, they are often in the same terminal, but due to the size of the alliances, it's not possible to keep them all together. So for Skyteam (for example), Delta international and KLM are in Terminal 4, but Air France and Alitalia are Terminal 1.
I don't know if there are airside jitneys connecting any terminals other than 2 and 4 (since Delta operates in both). As said, if you are changing terminals and there is no airside jitney, you have to go through security again. This is true even if you are all on one ticket, and even if the flights in question are in the same alliance (like my Delta to Air France example - per Delta's website, no jitney connects Terminal 2 or 4 to Terminal 1). This is one reason that changing planes at JFK can be a hassle; it was designed long before current airline security procedures.