  Moduli spaces of bounded local $G$-shtukas are a group-theoretic
generalization of the function field analog of Rapoport and Zink's moduli
spaces of $p$-divisible groups. In this article we generalize some very
prominent concepts in the theory of Rapoport-Zink spaces to our setting. More
precisely, we define period spaces, as well as the period map from a moduli
space of bounded local $G$-shtukas to the corresponding period space, and we
determine the image of the period map. Furthermore, we define a tower of
coverings of the generic fiber of the moduli space which is equipped with a
Hecke action and an action of a suitable automorphism group. Finally we
consider the $\ell$-adic cohomology of these towers.
