JFK Transportation Between Terminals
 

I have never been to JFK before so forgive the ignorance, but how do you get from terminal 4 (arriving from LAX) to terminal 1 (departing for Rome)? Will I need to go through security again? I have two hours between flights. Thanks in advance.

JFK Transportation Between Terminals
 

You'll have to take the AirTrain. It runs outside security, so yes. You'll have to reclear security. Unless there's a airside bus, I don't know about.

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I believe that the only air-side bus from T4 goes to T2 (for passengers connecting on Delta).

The OP will need to take the "All Terminals" AirTrain two stops (about a five-minute ride) to get from T4 to T1. Care should be taken not to get on a Jamaica- or Howard Beach-bound AirTrain, as each of those can take about 30 minutes to get from T4 to T1!

WITHIN the airport, the AIRTRAIN is FREE.^

Are you on separate tickets or on one routing? If the former we should alert you to the diners of missing your flight to Rome...

JFK Transportation Between Terminals
 

Two hours between flights while having to change terminals at JFK is very risky. If everything is on time, there's no lines at security, and you can run between flights you'll probably be ok, but it could just as easily go the other way. And, one thing to keep in mind if you are on separate tickets and have to recheck a bag, many airlines have 45-60 minute bag check cut offs. I've had short connections a JFK that I would have made but for missing the bag check cut off.

Hope your flights are on time...

More info would certainly be useful!

Most likely this is a DL>AZ connection, so most likely it's a single ticket.

I am on one routing: LAX to JFK on Delta, Friday Sept 9, arriving terminal 4 at 4:04 pm. Departing JFK terminal 1 for Rome on Alitalia at 6:05 pm. I am registered with TSA PRE check, for what that is worth as to going through security. We have carry-on luggage only. Is it walkable? Thanks for any additional information.

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Not walkable. Take the AirTrain, as described in Post # 3.

That's absurd. If the inbound flight is on time, this is a piece of cake.

I'd say it's pretty safe but not a piece of cake. Immigration could easily take 30-45 minutes and the non-Pre-check line could easily take another 30 minutes, and that's not including baggage claim and rechecking. If OP gets lost and takes the wrong AirTrain (note to OP: take the yellow line), plus somewhat typical long lines, there'd be no margin left.

Last I checked, one does not need to clear immigration and customs when arriving at JFK from LAX!

Sadly, nothing in this case as Alitalia does not participate and there is no PreCheck at JFK T1. Still, you should be fine with the 2 hour connection you have. Follow the advice above re AirTrain and do not dally, but you should be fine. The longer lines at T1 don't pick up until a bit later IME.