We study the impact of three-body physics in quenched unitary Bose gases, focusing on the role of the Efimov effect. Using a local density model, we solve the three-body problem and determine three-body decay rates at unitarity, finding density-dependent, log-periodic Efimov oscillations, violating the expected continuous scale invariance in the system. We find that the breakdown of continuous scale invariance, due to Efimov physics, manifests also in the earliest stages of evolution after the interaction quench to unitarity, where we find the growth of a substantial population of Efimov states for densities in which the interparticle distance is comparable to the size of an Efimov state. This agrees with the early-time dynamical growth of three-body correlations at unitarity [Colussi etal., Phys. Rev. Lett. 120, 100401 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.100401]. By varying the sweep rate away from unitarity, we also find a departure from the usual Landau-Zener analysis for state transfer when the system is allowed to evolve at unitarity and develop correlations.