The origin of life must have been preceded by Darwin-like evolutionary dynamics that could propagate it. How did that adaptive dynamics arise? And from what prebiotic molecules? Using evolutionary invasion analysis, we develop a universal framework for describing any origin story for evolutionary dynamics. We find that cooperative autocatalysts, i.e. autocatalysts whose per-unit reproductive rate grows as their population increases, have the special property of being able to cross a barrier that separates their initial degradation-dominated state from a growth-dominated state with evolutionary dynamics. For some model parameters, this leap to persistent propagation is likely, not rare. We apply this analysis to the Foldcat Mechanism , wherein peptides fold and help catalyse the elongation of each other. Foldcats are found to have cooperative autocatalysis and be capable of emergent evolutionary dynamics.