Green’s Dictionary of Slang

flap v.2

to rob, to swindle; thus flap the dimmock v., to pay (money).

[UK]Farmer & Henley Sl. and Its Analogues.

In phrases

flap a jay (v.) [jay n.1 (4)]

(UK Und.) to trick a simpleton, to swindle an innocent victim.

[UK]Daily Tel. 18 Aug. 3/1: He and three others of the division had cut up 70 between them, obtained by flapping a jay, which, rendered into intelligible English, means plundering a simple-minded person [F&H].
[US](con. 1930s) N. Algren ‘The Last Carousel’ Texas Stories (1995) 139: We moved the minches and flapped the jays every night.