  We use time-resolved nonlinear pump--probe measurements to reveal features of
semiconductor moir\'e materials not accessible to linear spectroscopy. With an
intense, red-detuned pump pulse, we generate a high density of virtual excitons
or exciton--polarons in various moir\'e minibands. A broadband probe pulse in
turn measures the response of all optical resonances induced by the
pump-generated excitations. We generically observe a coherent blue shift
originating from contact-like exciton--exciton interactions. At charge
neutrality, these measurements allow us to assess the spatial overlap between
different optical excitations and to observe signatures of a bound biexciton
state between two different moir\'e exciton modes. In contrast to electron
doped monolayers, spatially confined moir\'e attractive polarons behave as an
ensemble of non-interacting two-level emitters, exhibiting an
electron-density-independent ac-Stark effect. Tuning the pump laser into
resonance with the attractive polaron, we demonstrate the filling of the
moir\'e lattice with localized polarons and thereby realize a nonequilibrium
Bose--Fermi mixture in moir\'e flat bands.
