If you go to myairplane.com and look at the STARs coming into the New York area, you'll see they often tell pilots to "expect clearance" to cross so-and-so fix at altitude so-and-so. That clearance will come from the Center (New York Center is in Ronkonkoma), and typically Center will hand off the flight to Approach (in Westbury) as it's approaching that fix.
For instance, flights into JFK
from the west use the LENDY arrival and are cleared by NY Center to descend to FL190. As the flight is getting down toward that altitude, maybe 5-10 miles WNW of Paterson NJ
, Center tells them to contact Approach on 127.4 MHz. That's when they start talking to Westbury.
Flights out of NY call Westbury a minute or so after takeoff and mostly stay with them until they're up around 12000-15000 ft.