  A recently developed dynamic mean-field theory for disordered spins
(spinDMFT) is shown to capture the spin dynamics of nuclear spins very well.
The key quantities are the spin autocorrelations. In order to compute the free
induction decay (FID), pair correlations are needed in addition. They can be
computed on spin clusters of moderate size which are coupled to the dynamic
mean fields determined in a first step by spinDMFT. We dub this versatile
approach non-local spinDMFT (nl-spinDMFT). It is a particular asset of
nl-spinDMFT that one knows from where the contributions to the FID stem. We
illustrate the strengths of nl-spinDMFT in comparison to experimental data for
CaF$_2$. Furthermore, spinDMFT provides the dynamic mean fields explaining the
FID of the nuclear spins of $^{13}$C in adamantane up to some static noise. The
spin Hahn echo in adamantane is free from effects of static noise and agrees
excellently with the spinDMFT results without further fitting.
