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 φ 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 ≥ 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.