  Quenched randomness can have a dramatic effect on the dynamics of isolated 1D
quantum many-body systems, even for systems that thermalize. This is because
transport, entanglement, and operator spreading can be hindered by `Griffiths'
rare regions which locally resemble the many-body-localized phase and thus act
as weak links. We propose coarse-grained models for entanglement growth and for
the spreading of quantum operators in the presence of such weak links. We also
examine entanglement growth across a single weak link numerically. We show that
these weak links have a stronger effect on entanglement growth than previously
assumed: entanglement growth is sub-ballistic whenever such weak links have a
power-law probability distribution at low couplings, i.e. throughout the entire
thermal Griffiths phase. We argue that the probability distribution of the
entanglement entropy across a cut can be understood from a simple picture in
terms of a classical surface growth model. Surprisingly, the four length scales
associated with (i) production of entanglement, (ii) spreading of conserved
quantities, (iii) spreading of operators, and (iv) the width of the `front' of
a spreading operator, are characterized by dynamical exponents that in general
are all distinct. Our numerical analysis of entanglement growth between weakly
coupled systems may be of independent interest.
