  The scrambling of quantum information in closed many-body systems, as
measured by out-of-time-ordered correlation functions (OTOCs), has lately
received considerable attention. Recently, a hydrodynamical description of
OTOCs has emerged from considering random local circuits, aspects of which are
conjectured to be universal to ergodic many-body systems, even without
randomness. Here we extend this approach to systems with locally conserved
quantities (e.g., energy). We do this by considering local random unitary
circuits with a conserved U$(1)$ charge and argue, with numerical and
analytical evidence, that the presence of a conservation law slows relaxation
in both time ordered {\textit{and}} out-of-time-ordered correlation functions,
both can have a diffusively relaxing component or "hydrodynamic tail" at late
times. We verify the presence of such tails also in a deterministic,
peridocially driven system. We show that for OTOCs, the combination of
diffusive and ballistic components leads to a wave front with a specific,
asymmetric shape, decaying as a power law behind the front. These results also
explain existing numerical investigations in non-noisy ergodic systems with
energy conservation. Moreover, we consider OTOCs in Gibbs states, parametrized
by a chemical potential $\mu$, and apply perturbative arguments to show that
for $\mu\gg 1$ the ballistic front of information-spreading can only develop at
times exponentially large in $\mu$ -- with the information traveling
diffusively at earlier times. We also develop a new formalism for describing
OTOCs and operator spreading, which allows us to interpret the saturation of
OTOCs as a form of thermalization on the Hilbert space of operators.
