Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in three-dimensional $q$-state clock models, in which the ordered phase breaks the $Z_q$ discrete symmetry, while an emergent $\text{U}\left(1\right)$ symmetry appears at the critical point. The ${\mathrm{Z}}_5$-clock model with both isotropic and anisotropic couplings is taken as an example. By increasing the temperature at a finite velocity $v$ to traverse the critical point from the ordered phase, we find that in the driven process, there exist two finite-time scaling (FTS) regions, characterized by two driving-induced time scales ${\zeta }_d\propto v^{-z/r}$ and ${\zeta }_d^{\prime }\propto v^{-z/r^{\prime }}$, respectively. Here $z$ is the dynamic exponent, $r$ is the usual critical exponent of $v$, and $r^{\prime }$ represents an additional critical exponent of $v$ associated with the dangerously irrelevant scaling variable. While the square of the order parameter $M^2$ obeys the usual FTS form, the angular order parameter ${\varphi }_q$ shows remarkably distinct scaling behaviors controlled by both FTS regions. For small $v, {\varphi }_q$ is dominated by the time scale ${\zeta }_d$, whereas for large $v, {\varphi }_q$ is governed by the second time scale ${\zeta }_d^{\prime }$. These results confirm the universality of the dynamic scaling form proposed in Nat. Commun. 16, 3402 (2025), wherein only the isotropic ${\mathrm{Z}}_6$-clock model is considered. Besides, we find that the value of $r^{\prime }$ changes from the value of the isotropic case to another value as the strength of anisotropy increases. The crossover effect between weak and strong anisotropy is also explored. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal $R{\mathrm{MnO}}_3$ ($R$=rare earth) materials.

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• Critical phenomena
• Nonequilibrium statistical mechanics
• Simulated annealing