We get some rushes in our hay, since we have low and wet meadows. I’d say they compost slower than straw, since they have a much harder surface and more likely contain more carbon. But if you’re not in a hurry, you can just let them compost longer, the result will be the same in the end.
But I also think that they might not suck up the manure as well as straw, again owing to the hard surface. Maybe you can mix straw and rushes?