When introducing dynamics to classical scale-free random graphs, natural choices include considering stochastic processes such as the contact process on the (static) graph, introducing edge updating mechanisms to the graph itself, or studying both together and the resulting interactions. The latter has been studied extensively by Jacob, Linker and Mörters in a series of papers published over the past few years. In this talk we will look at some of the early results on introducing the contact process to scale-free geometric random graphs, highlighting a few of the issues that arise from the spatial embedding of the graphs. We will also see how a solution to one such difficulty can be used to study a dynamic version of these graphs, where vertices are allowed to move as independent Brownian motions and edges are regularly resampled. We will present some preliminary results about information propagation on such dynamic graphs, as well as some of the more general intermediate results that are used to study dynamic particle systems that drive the motion of vertices. This talk is based on joint works with Arne Grauer.