My kitties do this frequently too. Too frequently, in fact. Our neighborhood used to be filled with sweet bird songs in the morning, but now, it's dead silent. Come this spring, it will start all over again, I am afraid.
Mine have brought in a few rabbits, with their back skin missing, a baby mole, birds, snakes, a brand new baby squirrel, and I'm sure other critters I can't remember. They bring them in alive and finish them off in the house. Once, my kitchen was full of bird feathers everywhere. I was so freaked out, I couldn't disinfect enough! Another time, my mom was visiting, sitting on the toilet in the spare bathroom when she kept hearing this crunch, crunch, crunch. She threw open the shower curtain, and there was Pumpkin eating a bird! He had it in pieces all over the place. Oh, the stories! The horror stories! I could go on and on, but I won't. It's too heartbreaking. Murderers, every one!