I recently went through this when my passport was stolen in Peru. My immigration card was with my passport. When I got to the booth, I was directed to the supervisor who re-scanned my passport, searched and linked my entry record in the computer, and sold me a new immigration card for $5 US.
It was over in about 5 minutes and it was clear to me that it happens all the time. She didn't blink.
I flew to Argentina where they recognized the emergency passport right away and even stapled the sticker for my reciprocity fee into my emergency passport so that I could put in my new passport when I got home. I thought that was a nice touch. It all seemed very routine.
The biggest surprise was that I was able to update my Global Entry info online with the emergency passport number and use the kiosk without any special scrutiny at all on my return.