The flute has been part of klezmer ensembles for a very long time. There are some wonderful vintage recordings of bands where you can hear them playing heterophonically (doing different things with the melody) and adding trills and runs above the tune. We also have some great examples of solo playing.
Flutes played an important role taking the high melodic lines and fills in the early American bands of Art Shryer, Harry Kandel, and Abe Schwartz, to name but a few.
According to Adrianne Greenbaum, the world's foremost klezmer flautist, the instruments played in eastern European klezmer bands and in the US in the first part of the 1900s were generally of the multi-keyed variety, a few Boehm-system, but mostly simple-system and/or what we now sometimes refer to as 'transitional'.
Yiddish: fleyt, shtolper
Klezmer flautists in the late 19th and earlier 20th centuries included:
Argentina Leo Feidman, Sergio Feidman
USA Israel Chazin, Shloimke Kosch
Klezmer flautists in the later 20th century and today include:
France Marthe Desrosières (also clarinet)
USA Adrianne Greenbaum, Madeline Solomon