A few notes of info to help out:
Note 1: All rifle transactions (in this case) must be handled through an FFL.
Note 2: An FFL receiving a rifle must send a copy of their FFL to the seller and this paperwork must accompany the rifle during shipment.
Note 3: Somewhere in this transaction, the FFL has to get a copy of the buyers driver's license, but I can't remember exactly when that info exchange happens (before shipment, with shipment?? - I've done a few, but can't remember when I sent the DL info)
While there may be some logistical details that complicate the issue a little, it seems you would simply do this:
You sell your rifle (lower) to the NY person - their FFL sends you their FFL paperwork and you ship the rifle (lower) to their FFL.
They sell you their rifle (lower) - your FFL sends them their FFL paperwork and they ship their rifle (lower) to your FFL.
I say "rifle (lower)" because most grandfather clauses state that the rifle was supposed to be assembled before the ban took place. Just swapping lowers would technically be illegal, but proving that may be difficult.