Well, here's the scoop, fresh and hot from the bowels of the NYC reserve list:

LGA-JFK transfers.

Between 11 AM and 7 PM, the NYC express bus/van runs hourly, $13.60 one way with discount code "CREW." Install their app and Google their pickup spots, usually inside the terminal at the ground transportation desk. Talk to the person standing in the appointed spot in the hi-viz vest and let them scan your phone boarding pass and give you a printed ticket. This is apparently the successor of the NYC Airporter people have talked about.

You can also use the Kew Runner. Apparently 24 hours. The number the other guy posted is good for them. They're $18.50 direct to either airport, or $14.00 if you have the time, are coming from the airport itself, and are willing to sit in Kew Gardens to change vehicles.

Yellow Cabs are $55 flat rate between the airports generally. Uber & Lyft around $30 but varies, I've personally seen it as low as $17 (Lyft) and as high as $90 (Uber).