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Soft particulate gels can reversibly yield when sufficient deformation is applied, and the characteristics of this transition can be enhanced or limited by designing hybrid hydrogel composites. While the microscopic dynamics and macroscopic rheology of these systems have been studied separately in detail, the development of direct connections between the two has been difficult, particularly with regard to the nonlinear rheology. To bridge this gap, we perform a series of large amplitude oscillatory shear (LAOS) numerical measurements on model soft particulate gels at different volume fractions using coarse-grained molecular dynamics simulations. We first study a particulate network with local bending stiffness and then we combine it with a second component that can provide additional cross-linking to obtain two-component networks. Through the sequence of physical processes (SPP) framework, we define time-resolved dynamic moduli, and by tracking the changes in these moduli through the period, we can distinguish transitions in the material behavior as a function of time. This approach helps us establish the microscopic origin of the nonlinear rheology by connecting the changes in dynamic moduli to the corresponding microstructural changes during the deformation including the nonaffine displacement of particles, and the breakage, formation, and orientation of bonds.