Agriculture appeared first in the Fertile Crescent, around 12,000 years ago.

From reading in this forum, I realize that the current European population has little to do with the people living in Europe 12,000 years ago: we are a mixture of ANE, WHG and EEF. And this mixture happened relatively recently, at least with ANE.

So, my question is: is it known who these people at the Fertile Crescent were? I assume, with analogy to Europe, that those people do not exist anymore, at least in "pure" terms. But is it known who they were? Which characteristics did they have?