We consider the fate of 1/N expansions when a many-body quantum system is quenched across criticality and show the emergence of exp(2λt)/N as renormalized parameter ruling the quantum-classical transition and accounting non-perturbatively for the local divergence rate λ of mean-field solutions. In terms of exp(2λt)/N, quasiclassical expansions of paradigmatic examples of criticality like the selftrapping transition in an integrable Bose-Hubbard dimer and the generic instability of atractive bosonic systems towards soliton formation, are pushed to arbitrarily high orders. The agreement with numerical simulations supports the general nature of our results in the appropriately combined long-time λt → ∞ quasiclassical N → ∞ regime, out of reach of expansions in the bare parameter 1/N. For scrambling in many-body hyperbolic systems, our results provide formal grounds to a conjectured multi-exponential form of out-of-time-ordered correlators.