Critical relaxational dynamics at the continuous transitions of three-dimensional spin models with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mat

We characterize the dynamic universality classes of a relaxational dynamics under equilibrium conditions at the continuous transitions of three-dimensional (3D) spin systems with ${\mathbb{Z}}_{2}$ gauge symmetry. In particular, we consider the pure lattice ${\mathbb{Z}}_{2}$ gauge model and the lattice ${\mathbb{Z}}_{2}$ gauge XY model, which present various types of transitions: topological transitions without a local order parameter and transitions characterized by both gauge-invariant and non-gauge-invariant XY order parameters. We consider a standard relaxational (locally reversible) Metropolis dynamics and determine the dynamic critical exponent $z$ that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological ${\mathbb{Z}}_{2}$ gauge transitions we find $z=2.55(6)$. Therefore, the dynamics is significantly slower than in Ising systems---$z\ensuremath{\approx}2.02$ for the 3D Ising universality class---although 3D ${\mathbb{Z}}_{2}$ gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D ${\mathbb{Z}}_{2}$ gauge XY model, we find that their critical dynamics belong to the same dynamic universality class as the relaxational dynamics in ungauged XY systems independently of the gauge-invariant or non-gauge-invariant nature of the order parameter at the transition.