Oh dear. We once had a cat who did this. I used to have to throw a towel over the poor birds and pick them up using that. Another cat used to bring us plastic bread wrappers. I could count on finding one on the doorstep each morning. Daft thing he was. One of the others used to bring me his treasures in the night. Its amazing how crackly huge moths are when placed on one's pillow.
No more of such nonsense these days. Only one (huge) cat who is largely inert. Luckily he's very devoted and mostly spends his time as a cuddly, purring, furry, jelly on the sofa next to me.