The stack of $G$-zips is a Mori dream space

We first extend previous results of the author with T. Wedhorn and W. Goldring regarding the existence of $\mu$-ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of $G$-zips of Pink--Wedhorn--Ziegler. Then, we define and study the Cox ring of the stack of $G$-zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of $G$-zips and study its properties. We prove that it is finitely generated in certain cases.