  The real-time dynamics of equal-site correlation functions is studied for
one-dimensional spin models with quenched disorder. Focusing on infinite
temperature, we present a comparison between the dynamics of models with
different quantum numbers $S = 1/2, 1, 3/2$, as well as of chains consisting of
classical spins. Based on this comparison as well as by analyzing the
statistics of energy-level spacings, we show that the putative many-body
localization transition is shifted to considerably stronger values of disorder
for increasing $S$. In this context, we introduce an effective disorder
strength $W_\text{eff}$, which provides a mapping between the dynamics for
different spin quantum numbers. For small $W_\text{eff}$, we show that the
real-time correlations become essentially independent of $S$, and are moreover
very well captured by the dynamics of classical spins. Especially for $S =
3/2$, the agreement between quantum and classical dynamics is remarkably
observed even for very strong values of disorder. This behavior also reflects
itself in the corresponding spectral functions, which are obtained via a
Fourier transform from the time to the frequency domain. As an aside, we also
comment on the self-averaging properties of the correlation function at weak
and strong disorder. Our work sheds light on the correspondence between quantum
and classical dynamics at high temperatures and extends our understanding of
the dynamics in disordered spin chains beyond the well-studied case of $S=1/2$.
