I would say ( former Ny resident here) if you fly into a NY airport outside of the NYC metro region AND the gun in question has no issues with any of the myriad of confusing NY state “assault weapon “ regulations, no one will bat an eye.

Your typical manually operated pump bolt lever type will be completely fine.

Try to walk the tight rope of the “assault weapons” regulations, and that is where trouble ensures, even if the item in question is compliant, as the vast majority of enforcement officials have FAR LESS knowledge about this topic than almost anyone on these forums, and the typical response is detain / arrest/ confiscate now ask questions later.

So to put it simply if you are traveling with a manually operated long gun no worries

Same goes for a “ conventional “ appearing semi like a ruger mini 14 and 10 round or less magazines

If however you are transporting an AR, even a fixed mag or features less, be prepared to potentially get jacked up

Been in and out of the state with a plain jane 10/22 in WNY and Albany airports. No problem. 10 round standard mags.

Since Hochul had her fit after Bruen ?

Section 265.01-e Criminal possession of a firearm, rifle or shotgun in a sensitive location.

1. A person is guilty of criminal possession of a firearm, rifle or shotgun in a sensitive location when such person possesses a firearm, rifle or shotgun in or upon a sensitive location, and such person knows or reasonably should know such location is a sensitive location.
   
2. For the purposes of this section, a sensitive location shall mean:

(n) any place, conveyance, or vehicle used for public transportation or public transit, subway cars, train cars, buses, ferries, railroad, omnibus, marine or aviation transportation; or any facility used for or in connection with service in the transportation of passengers, airports, train stations, subway and rail stations, and bus terminals;

Criminal possession of a firearm, rifle or shotgun in a sensitive location is a class E felony.

Yes. And the ATF said they wouldn't go door-to-door for forced reset triggers.

We are worse off now than before Bruen.

Such a shitty situation.

Seems to be a case of taking a step back to make many leaps forward. I am still confident that NY will get slapped down, but yes things are worse at the moment.

All the forms for Rifle/shotgun permit for NYC say you can carry on a train if locked in case.  So that is out the window now?

Cannot carry a concealed Handgun into a "Sensitive location" unless permission has been granted fist.
Walking into an Airport with a cased (unloaded) Long Gun or Handgun isn't going to be an issue as ostensibly it's already packaged for transportation. Outbound, you make the declarations to the Airline Carrier Counter, they give the high sign to TSA and they inspect the gun and ammunition, lock it, and send it through Baggage.  Handguns: Generally TSA contacts local LEO to verify a valid NY Handgun Licenses and that the handgun being inspected is on the Licenses and you're good to go.
Inbound: Last few times I carried, a Baggage Handler met me at the pick up area, verified it was my Bag and delivered it in person.. ymmv

Penal Law Section 265.01-e Criminal possession of a firearm, rifle or shotgun in a sensitive location.
 
1. A person is guilty of criminal possession of a firearm, rifle or shotgun in a sensitive location when such person possesses a firearm, rifle or shotgun in or upon a sensitive location, and such person knows or reasonably should know such location is a sensitive location.

My son and his friend to and back from Alaska last fall. No issues. As a matter of fact the cases came out with the  baggage and they just picked them up and left. This was in Rochester whatever its called now airport.