We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of Complex Quantum Network Manifolds in dimensions $d > 2$, and special types of Network Geometry with Flavour models studied in the physics literature by Bianconi, Rahmede [$\mathit{Sci. Rep.} \; \mathbf{5},\text{ 13979 (2015) and }\mathit{Phys. Rev. E} \; \mathbf{93},\text{ 032315 (2016)}$].