The "mixing with local cultures" is a process that took centuries, and in cultures already connected to similar tenets and beliefs.
It was not a case of "I saw a documentary/bought a manual on budhism, I am so zen now" while shopping stuff from Whole Foods.
Wha? Buddhism has a long history of evolving and mixing with local cultures, just like any religion. The zen stuff that these "koans" are either imitating or satirizing (depending on your view) comes from mixing Buddhism and Confucianism and Taoism and who knows what else.