I will discuss the “scrambling” of local quantum information in chaotic quantum many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a dissipative process. Our results are obtained by examining the dynamics of operator spreading under unitary time evolution in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator spreads ballistically with a front that moves at a “butterfly speed”, but develops a power law “tail” behind its leading ballistic front due to the slow dynamics of the conserved component of the operator. This structure implies that the out-of-time-order commutator (OTOC) between two initially spatially separated operators grows sharply upon the arrival of the ballistic front but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. I will also present these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.