Why do we say 'flea market'?

March aux puces - original flea market still popular in Paris

A mysterious term given that the market for fleas is limited. There are two (vaguely) plausible theories:

1. March aux puces?

A translation of march aux puces. This was the popular name for a large outdoor market in Paris that became popular in the 1920s. 

The name march aux puces was "because there are so many second hand articles sold of all kinds that they are believed to gather fleas." [E.S. Dougherty, "In Europe," 1922] 

2. Dutch Swamp?

From the Dutch word for swamp is given as “vlie”, which sounds like flea when spoken in English. The Dutch settlers held markets in the then swampland that was Manhattan Island. 

The OED goes with the French market explanation as the more probable.