  Charge order is universal among high-T$_c$ cuprates but its relevance to
superconductivity is not established. It is widely believed that, while static
order competes with superconductivity, dynamic order may be favorable and even
contribute to Cooper pairing. We use time-resolved resonant soft x-ray
scattering to study the collective dynamics of the charge order in the
prototypical cuprate, La$_{2-x}$Ba$_x$CuO$_4$. We find that, at energy scales
$0.4$ meV $ \lesssim \omega \lesssim 2$ meV, the excitations are overdamped and
propagate via Brownian-like diffusion. At energy scales below 0.4 meV the
charge order exhibits dynamic critical scaling, displaying universal behavior
arising from propagation of topological defects. Our study implies that charge
order is dynamic, so may participate tangibly in superconductivity.
