The translocation dynamics of cells and particles through geometric constrictions are critical in biological and biomedical processes from splenic filtration to tumor metastasis. While particle stiffness plays a key role, its role in highly nonequilibrium states remains poorly understood. Here, we present a multiscale model to investigate the impact of particle stiffness on the translocation dynamics in microfluidic channels. We find that semielastic particles exhibit superior translocation capabilities compared to both softer and more rigid particles, with a nonmonotonic stiffness dependence observed for highly deformable particles. Additionally, we identify crossover behaviors in translocation time driven by variations in the flow rate, particle size, and particle-plate interactions. Excessive particle deformation significantly regulates these dynamics, with stiffness-induced shape transitions from pancake-like to ellipsoidal forms, controlling frictional forces at the particle-channel interface and the sieve plate. The balance between these forces explains the observed nonmonotonic translocation dynamics. Our work provides insights into the relationship between particle deformability and flow dynamics, highlighting the importance of elasticity in translocation behavior. These findings have implications for designing microfluidic devices for efficient separation and analysis of cells with varying elasticities, advancing applications in human health diagnostics.