  We consider the fate of $1/N$ expansions in unstable many-body quantum
systems, as realized by a quench across criticality, and show the emergence of
${\rm e}^{2\lambda t}/N$ as a renormalized parameter ruling the
quantum-classical transition and accounting nonperturbatively for the local
divergence rate $\lambda$ of mean-field solutions. In terms of ${\rm
e}^{2\lambda t}/N$, quasiclassical expansions of paradigmatic examples of
criticality, like the self-trapping transition in an integrable Bose-Hubbard
dimer and the generic instability of attractive bosonic systems toward 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 $\lambda t\to \infty$ quasiclassical $N\to \infty$ 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 multiexponential form of out-of-time-ordered correlators.
