The efficiency of solution-processed colloidal quantum dot (QD) based solar cells is limited by poor charge transport in the active layer of the device, which originates from multiple trapping sites provided by QD surface defects. We apply a recently developed ultrafast electro-optical technique, pump-push photocurrent spectroscopy, to elucidate the charge trapping dynamics in PbS colloidal-QD photovoltaic devices at working conditions. We show that IR photoinduced absorption of QD in the 0.2-05 eV region is partly associated with immobile charges, which can be optically detrapped in our experiment. Using this absorption as a probe, we observe that the early trapping dynamics strongly depend on the nature of the ligands used for QD passivation, while it depends only slightly on the nature of the electron-accepting layer. We find that weakly bound states, with a photon-activation energy of 0.2 eV, are populated instantaneously upon photoexcitation. This indicates that the photogenerated states show an intrinsically bound-state character, arguably similar to charge-transfer states formation in organic photovoltaic materials. Sequential population of deeper traps (activation energy 03-05 eV) is observed on the similar to 0.1-10 ns time scales, indicating that most of carrier trapping occurs only after substantial charge relaxation/transport. The reported study disentangles fundamentally different contributions to charge trapping dynamics in the nanocrystal-based optoelectronic devices and can serve as a useful tool for QD solar cell development.