  Quantum spin liquids are topological states of matter that arise in
frustrated quantum magnets at low temperatures. At low energies, such states
exhibit emergent gauge fields and fractionalized quasiparticles and can also
possess enhanced global symmetries compared to their parent microscopic
Hamiltonians. We study the consequences of this emergent gauge and symmetry
structure for the hydrodynamics of quantum spin liquids. Specifically, we
analyze two cases, the $U(1)$ spin liquid with a Fermi surface and the
$SU(4)$-symmetric "algebraic" spin liquid. We show that the emergent degrees of
freedom in the spin liquid phase lead to a variety of additional hydrodynamic
modes compared to the high-temperature paramagnetic phase. We identify a
hydrodynamic regime for the internal $U(1)$ gauge field common to both states,
characterized by slow diffusion of the internal transverse photon.
