  Understanding fluctuation phenomena plays a dominant role in the development
of many-body physics. The time evolution of entanglement is essential to a
broad range of subjects in many-body physics, ranging from exotic quantum
matter to quantum thermalization. Stemming from various dynamical processes of
information, fluctuations in entanglement evolution differ conceptually from
out-of-equilibrium fluctuations of traditional physical quantities. Their
studies remain elusive. Here we uncover an emergent random structure in the
evolution of the many-body wavefunction in two classes of integrable -- either
interacting or noninteracting -- lattice models. It gives rise to
out-of-equilibrium entanglement fluctuations which fall into the paradigm of
mesoscopic fluctuations of wave interference origin. Specifically, the
entanglement entropy variance obeys a universal scaling law, in each class, and
the full distribution displays a sub-Gaussian upper and a sub-Gamma lower tail.
These statistics are independent of both the system's microscopic details and
the choice of entanglement probes, and broaden the class of mesoscopic
universalities. They have practical implications for controlling entanglement
in mesoscopic devices.
