A few loads will probably not last very long, but absolutely, give those girls some yard "waste", they will love it!
When there isn't any bulk
carbon in the run, I will turn the soil with a fork.
They come running, picking through the chunks, scratching, digging, eating things too small for me to see.
I even give them woody branch prunings that have green leaves.
The eat what they want and leave the rest.
I leave it until I'm ready to turn the soil, then I rake them into a pile for bio char making.