  In a number of physical situations, from polarons to Dirac liquids and to
non-Fermi liquids, one encounters the "beyond quasiparticles" regime, in which
the inelastic scattering rate exceeds the thermal energy of quasiparticles.
Transport in this regime cannot be described by the kinetic equation. We employ
the Diagrammatic Monte Carlo method to study the mobility of a Fr\"{o}hlich
polaron in this regime and discover a number of non-perturbative effects: a
strong violation of the Mott-Ioffe-Regel criterion at intermediate and strong
couplings, a mobility minimum at $T \sim \Omega$ in the strong-coupling limit
($\Omega$ is the optical mode frequency), a substantial delay in the onset of
an exponential dependence of the mobility for $T<\Omega$ at intermediate
coupling, and complete smearing of the Drude peak at strong coupling. These
effects should be taken into account when interpreting mobility data in
materials with strong electron-phonon coupling.
