Good morning everyone.
As a person who spends most of their time on the PJ branch since It's the closest to me. I can say the following.
Service to Kleet Lumber via New York & Atlantic Railway is not often and as needed. Like mentioned, they will drop the cars at Kleet where the Forklifts take it from there, then they head east to Northport and runaround the train in the siding just east of the station. Interlocking titles DUKE1 and DUKE2.