Numerous dynamical systems describing real world phenomena exhibit a characteristic fine structure which means that they are composed of smaller subsystems interacting with each other. This interaction structure can be represented by a coupling network whereby the original system can be viewed as a network of dynamical systems and is commonly termed coupled cell system. Since reality crucially depends on time, derived models generally tend to be subject to temporal changes as well. Particularly in applications involving technology, this temporal evolution often occurs as a consequence of an instantaneously varying network structure; communication networks provide a prominent class of examples. In this thesis, time-varying dynamical system networks are analyzed on the grounds of the following two structural aspects: Firstly, instantaneous modifications of the underlying coupling network generally lead to non-smooth vector fields and, secondly, a fixed network structure naturally introduces symmetries to the according system. These analytical and algebraic observations trigger the system's description as a hybrid dynamical system with local symmetry information. In search of global structure for systems of such kind, a global symmetry framework for hybrid dynamical systems formulated in terms of hybrid automata is unfolded that takes into account both discrete transition graph symmetries and local dynamical systems' symmetries giving rise to the concept of hybrid symmetries. Restricted to a special class of switched systems which induce hybrid automata by the choice of a switching signal, symmetry-induced switching strategies termed orbital switching signals are investigated and stability issues of switched linear systems are addressed. Against this theoretical background, examples of time-varying dynamical system networks are treated both structurally and numerically for orbitally switched coupling networks.