We investigate femtosecond laser trapping dynamics of two-photon absorbing hollow-core nanoparticles with different volume fractions and two-photon absorption (TPA) coefficients. Numerical simulations show that the hollow-core particles with low and high-volume fractions can easily be trapped and bounced by the tightly focused Gaussian laser pulses, respectively. Further studies show that the hollow-core particles with and without TPA can be identified, because the TPA effect enhances the radiation force, and subsequently the longitudinal force destabilizes the trap by pushing the particle away from the focal point. The results may find direct applications in particle sorting and characterizing the TPA coefficient of single nanoparticles.