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In this paper, the dynamic of inertial capsules into microfluidic bifurcations is studied. The fluid evolution is based on the solution of the BGK -- lattice Boltzmann scheme including a forcing term accounting for immersed geometries. The dynamic-Immersed Boundary forcing strategy is adopted for imposing no-slip boundary conditions on moving deformable or rigid structures, while, on fixed immersed geometries the Bouzidi-Firdaouss-Lallemand second-order bounce back technique is implemented. The proposed computational framework is employed to detail dynamics and deformation of rigid and deformable capsules traveling into a branching duct. This journey is characterized in terms of i) the capsule/bifurcation interaction depending on the sharpness of the branching channels junction; ii) daughter branches aperture angle; iii) occlusion ratio, the ratio between capsule size and main channel diameter; iv) flowing capsules stiffness; v) number of flowing particles.