In this talk, I will present recent results on the contact process on two specific types of scale-free, inhomogeneous random networks that evolve either through edge resampling or by resampling entire neighborhoods of vertices. Depending on the type of graph, the selected stationary dynamic, the tail exponent of the degree distribution, and the updating rate, we identify parameter regimes that result in either fast or slow extinction. In the latter case, we determine metastable exponents that exhibit first-order phase transitions. This is joint work with Emmanuel Jacob (ENS Lyon) and Peter Mörters (Universität zu Köln).