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 Fröhlich polaron in this regime and discover a number of nonperturbative effects: a strong violation of the Mott-Ioffe-Regel criterion at intermediate and strong couplings, a mobility minimum at T∼Ω in the strong-coupling limit (Ω is the optical mode frequency), a substantial delay in the onset of an exponential dependence of the mobility for T<Ω 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.