There is no published literature on this particular case of lexical variation.
There is a clear North-South divide with respect to this case of lexical variation, where knock and run and other related terms (e.g. knock-a-door run, knock knock run) are favoured in the North and Midlands, and knock knock ginger and other members of the ginger family are favoured in the South. Two other terms, though they appear much less frequently, also show clear regional patterning: cherry knocking is clustered in a southern region of the Midlands, while nicky nicky nine doors seems to be restricted to the North East.
ditch family = ding dong ditch, doorbell ditch
ginger family = knock knock ginger, knock down ginger etc.
run family = knock and run, knock knock run, knock-a-door run etc.