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.