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 $mathcal{O}(0.5-12hbox{ fs})$, placing them on the brink of detection for modern time-resolved probes.
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