Thermal-gradient induced transport of ionic liquid (IL) and water droplets through a carbon nanotube (CNT) is investigated in this study using molecular dynamics simulations. Energetic analysis indicates that IL transport through a CNT is driven primarily by the fluid-solid interaction, while fluid-fluid interactions dominate in water-CNT systems. Droplet diffusion analysis via the moment scaling spectrum reveals sub-diffusive motion of the IL droplet, in contrast to the self-diffusive motion of the water droplet. The Soret coefficient and energetic analysis of the systems suggest that the CNT shows more affinity for interaction with IL than with the water droplet. Thermophoretic transport of IL is shown to be feasible, which can create new opportunities in nanofluidic applications.