Many processes in nature and technology can be modeled as networks of interacting oscillatory processes. In many cases, the underlying network structure is not static but evolves over time. Specifically, the network dynamics and the dynamics of the underlying network can interact. In this talk we look at two specific examples of dynamics on evolving networks. First, we look at network evolution that is fast compared to the network dynamics leading to a state-dependent network: For example, neural oscillators can be insensitive to new input in certain states, leading to effective 'deactivation' of network connections. Second, we look at network evolution that is slow compare to the network dynamics. Specifically, we look at damage processes in neurodegenerative diseases such as Alzheimer's disease and how they interact with the network dynamics as the disease evolves.