Best advice I can give , if traveling to NY, "Leave the gun, take the canoli!"...
The Federal 1986 FOPA law does protect you, to a point. The point is, you have to be travelling THROUGH the state. You are allowed to stop for gas, get a bite to eat, fix a flat, etc., but if your destination is in NY state, then no, you are not covered by the Federal law, state law has jurisdiction.
I do not believe NY recognizes any other state's permits and NYC does not even recognize NY STATE permits.
And the FOPA (assuming you were covered) does not stop you from getting arrested, having your gun confiscated, booked, and spending jail time until you make bail. AND you will have to come back to NY for your court hearing, possibly months later. It has happened.
DO NOT THINK OF TAKING A HANDGUN, or your pet tacticool AR or anything else that might remotely be thought of as an assault weapon. Your granddaddy's duck gun might be allowed, but you need to find out FIRST, and in writing, from a state official, who has the authority to make that decision.
I grew up in northern NY state. Left for good in the 1970s. The laws were bad then, and are much worse now.
Essentially, if you are physically within the borders of NY state, with a firearm, and you do not have the proper NYS paperwork for you and it, you are breaking the law. In NYC, it has to be NYC issued paperwork, or you are breaking the law. DON'T DO THAT!
ALL handguns require a permit, and NY state (and city) are not "shall issue" places, they are "may issue", which means it is entirely up to the issuing individual, and you can be denied for any, or no reason, at their whim. And that's for residents!
GOOD LUCK, hope this helps. There are "loopholes" I believe, but in general this is how it works. If you HAVE to take a gun to NY, contact a NY lawyer for advice, BEFORE you set foot in NY!
NJ, MA, and some other places in that area are very similar. One fellow, flying with guns checked as baggage, stopover in NJ. All good, until his plane had trouble, and he couldn't leave until the next day. Airline put him up in a hotel, but when he claimed his bags, he broke NJ law, and was arrested, spent a couple days in jail until his hearing, and then had to return to NJ, months later for his hearing to finally end the matter.
If you are covered by the FOPA, you will ultimately not be convicted, but that's about all. The money you have to spend, and the time taken from your life, until they officially absolve you is not something you will get back.