  We introduce a simple criterion for lattice models to predict quantitatively
the crossover between the classical and the quantum scaling of the Kibble-Zurek
mechanism, as the one observed in a quantum $\phi^4$-model on a 1D lattice
[Phys. Rev. Lett. 116, 225701 (2016)]. We corroborate that the crossover is a
general feature of critical models on a lattice, by testing our paradigm on the
quantum Ising model in transverse field for arbitrary spin-$s$ ($s \geq 1/2$)
in one spatial dimension. By means of tensor network methods, we fully
characterize the equilibrium properties of this model, and locate the quantum
critical regions via our dynamical Ginzburg criterion. We numerically simulate
the Kibble-Zurek quench dynamics and show the validity of our picture, also
according to finite-time scaling analysis.
