There's a short illustrated article on Lullingstone station here :
Lullingstone Station (kentrail.org.uk)
It's here on current Google Maps - you can see the serrated edge where the platform edge stones have been removed, as in the first photo above
Google Maps
The airport was to spread south-west from there, across what are still open fields to and beyond where the M25 motorway now is. In the 1930s, of course, the bulk of what airline traffic there was came from Paris/Brussels/Amsterdam etc, coming from this direction towards London. The project was one of the Southern Railway's, who bought the land, got the parliamentary approvals, and who were a significant shareholder in Imperial Airways, the pre-WW2 predecessor of British Airways.
This was an era when London was prone to fog/smog at its low level, and up on the plain of the North Downs, as here, was to get away from that.