We define an evolving Shelah‐Spencer process as one by which a random graph grows, with at each time a new node incorporated and attached to each previous node with probability, where is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah‐Spencer sparse random graphs discussed in [21] and throughout the model‐theoretic literature. The first order axiomatisation for classical Shelah‐Spencer graphs comprises a Generic Extension axiom scheme and a No Dense Subgraphs axiom (...) scheme. We show that in our context Generic Extension continues to hold. While No Dense Subgraphs fails, a weaker Few Rigid Subgraphs property holds. (shrink)