Polycrystalline Ni-based superalloys find use in aero-engine components. Their superior mechanical properties are owed to the composite microstructure, which consists of γ’ precipitates in the matrix of the γ phase. However, their microstructure-mechanical property correlations are still not fully understood. We develop a physically-based crystal plasticity constitutive model for studying thermomechanical deformation in these alloys. The framework has consideration for the underlying microscale deformation phenomena. An Arrhenius type activation energy-based constitutive equation is used for modeling thermally-activated dislocation glide. Slip system-level back stress is used as an internal state variable to track the directional hardening during cyclic deformation. Precipitate size-dependent strengthening mechanisms due to weak pair coupling, strong pair coupling and Orowan looping are considered in the framework. Further, a micromechanical model is developed to account for the experimentally observed precipitate shearing during cyclic deformation. Application of the model is demonstrated for a powder metallurgy-processed, polycrystalline superalloy, which has γ’ precipitates of three different sizes. Here, the secondary and tertiary γ’ precipitates provide strengthening at the sub-micron scale, while the primary γ’ precipitates and grain boundaries and annealing twins influence microscale deformation (and eventual failure) in the polycrystalline ensembles. The model is first used to predict misorientation developments during tensile deformation and compared with the experimental EBSD counterparts. Based on the analysis, failure initiators are identified in these polycrystalline microstructures. Finally, the model is used to predict cyclic deformation and associated microstructure evolution due to precipitate shearing. Model predictions of the different strengthening contributions indicate that precipitate shearing is the primary contributor to cyclic softening observed in these superalloys.