If you look at the map, there is no place to stage it. The border with Turkey is very small. Staging in Iraq would mean troops under attack from both sides even before they cross the border.

The Kurds soundly control a mountain land corridor between Iraq and Iran, and have controlled that since even before the recent attacks. Iran is not in a position anytime soon to close that without air power that they don't have.

I don't think staging light infantry will be a problem but I don't think they'll successfully break out of the mountains, certainly not into any land that isn't already ethnically Kurd (or Baloch, but that's in the other corner of the country).