We present numerical simulations of a particle trapped at the isotropic-nematic liquid crystal (Iso-N) interface. We use our recent model, based on a phase-field approach [see Qiu et al., Phys. Rev. E 103, 022706 (2021)], to couple the capillary forces acting on the interface with the elastic stresses in the nematic phase along with topological defects. A range of floating configurations are first investigated as a function of the contact angle and various anchoring conditions at the fluid interface. The results show that the response of the system is driven by the existence of an anchoring conflict at the contact line. Substantial particle displacements and/or interfacial deformations may occur in this case even for moderate anchoring strengths. These findings highlight the coupling between elastic and capillary forces. In a second part, we compute drag forces exerted on a particle that moves along the Iso-N interface for several contact angles and a moderate Ericksen number. Because of the coupling between the velocity and order parameter fields, topological defects are swept downstream of the particle by the flow and sometimes escape from the particle or merge with the interface. We also find linear force-velocity laws, with drag forces at the Iso-N interface being slightly greater than their isotropic counterparts due to director distortions. We discuss these results in light of past studies on the behavior of particles being dragged in the bulk of a liquid crystal matrix.Read less <