Without getting into all the specifics, ATC is supposed to inform jets and large props when they leave Class B airspace and reentry if they cannot keep them inside Class B airspace while being vetored for an approach.
As far as the VOR-DME-A approach is concerned X WAYNES at 3000 decsending to X JAYMO at 2500 and CLIFO at 2000 will put you into CDW Class D airspace. The final approach course appears to be contained within the extened vertical limits of CDW Class D and into the Class B. Descent to CLIFO goes through Class D and directly into Class B.
I don’t know the local specifis but at Potomac TRACON there is a similar situation with Manassas Class D and the IAPs to the 1’s at IAD. PCT has a letter of agreement inplace with HEF that caps the HEF Class D authority at 2000 feet. If an aircraft should call to transit the CLASS D at 2500 feet the tower would direct the aircraft to call the TRACON so that radar services and traffic advisories could be provided.
The situation looks the same and I would bet that the NY TRACON cut the same deal with CDW.
When the tower is closed Class E rules would apply.
Hope this helps!