Mathematical models to describe the spread of particles in a structured population or network are highly relevant in multiple disciplines such as biology, physics and computer science. To understand the course of an epidemic, the spread of a computer virus in a network or the dissemination of misinformation in social media are crucial challenges in modern society. These processes heavily depend on the spatial structure, which is often described by a graph. An effect which is often neglected, is that in reality this structure is not fixed and evolves as time progresses. Thus, it is a natural choice to consider interacting particle systems on dynamical graphs. Our focus lies on the contact process, which is a benchmark for models of the spread of an infection in a structured population. The overarching goal of this proposal is to gain more insight about the contact process on dynamical graphs. We formulate three concrete objectives. In two of them the underlying graph is an infinite transitive graphs with bounded degree. The goal of the first objective is to find the asymptotic shape and the expansion speed of the infection region and the second objective is a better understanding of immunization effects that may occur in these models. Here immunization means that the dynamical graph structure forces the infection to go extinct no matter how high the infection rate is. The third objective deals with finite graphs. Our aim is to understand the metastable behaviour of a contact process on a dynamical configuration model with a power law degree distribution and identify the influence of the dynamical graph structure.