  Our theoretical understanding of heavy-fermion compounds mainly derives from
iconic models, such as those due to Kondo or Anderson. While providing
invaluable qualitative insight, detailed comparisons to experiments are
encumbered by the materials' complexity, including the spin-orbit coupling,
crystal fields, and ligand hybridizations. Here, we study the paradigmatic
Kondo insulator Ce$_3$Bi$_4$Pt$_3$ with a first principles dynamical mean-field
method that includes these complications. We find that salient signatures of
many-body effects in this material---large effective masses, the
insulator-to-metal crossover, the concomitant emergence of Curie-Weiss
behaviour and notable transfers of optical spectral weight---are captured
quantitatively. With this validation, we elucidate the fabric of the many-body
state. In particular, we extent the phenomenology of the Kondo crossover to
time-scales of fluctuations: We evidence that spin and charge degrees of
freedom each realize two regimes in which fluctuations adhere to vastly
different decay laws. We find these regimes to be separated by a {\it common}
temperature $T^{max}_\chi$, linked to the onset of Kondo screening.
Interestingly, below (above) $T^{max}_\chi$, valence fluctuations become faster
(slower) than the dynamical screening of the local moments. Overall, however,
spin and charge fluctuations occur on comparable time-scales of
$\mathcal{O}(0.5-12\hbox{ fs})$, placing them on the brink of detection for
modern time-resolved probes.
