Composting

It’s not drying that causes leaves to go brown. The plant actively pumps its waste products into leaves which turns them brown. Also some leaves can oxidise when damaged.

Dense mats of grass won't compost well, people with large lawns get this sometimes.
It'll do a lot better if you mix in other stuff (kitchen scraps, hedge trimmings, coffee grounds).
Mix the top of your heap with your pitchfork after adding grass, or actually turn completely but that's more work.