  In materials with strong electron-phonon ($e$-ph) interactions, the electrons
carry a phonon cloud during their motion, forming quasiparticles known as
polarons. Predicting charge transport and its temperature dependence in the
polaron regime remains an open challenge. Here, we present first-principles
calculations of charge transport in a prototypical material with large
polarons, SrTiO$_{3}$. Using a cumulant diagram-resummation technique that can
capture the strong $e$-ph interactions, our calculations can accurately predict
the experimental electron mobility in SrTiO$_{3}$ between 150$-$300 K. They
further reveal that for increasing temperature the charge transport mechanism
transitions from band-like conduction, in which the scattering of renormalized
quasiparticles is dominant, to a beyond-quasiparticle transport regime governed
by incoherent contributions due to the interactions between the electrons and
their phonon cloud. Our work reveals long-sought microscopic details of charge
transport in SrTiO$_{3}$, and provides a broadly applicable method for
predicting charge transport in materials with strong $e$-ph interactions and
polarons.
