Oceanic mesoscale eddies contribute important horizontal heat and salt transports on a global scale. Here we show that eddy transports are mainly due to individual eddy movements. Theoretical and observational analyses indicate that cyclonic and anticyclonic eddies move westwards, and they also move polewards and equatorwards, respectively, owing to the β of Earth's rotation. Temperature and salinity (T/S) anomalies inside individual eddies tend to move with eddies because of advective trapping of interior water parcels, so eddy movement causes heat and salt transports. Satellite altimeter sea surface height anomaly data are used to track individual eddies, and vertical profiles from co-located Argo floats are used to calculate T/S anomalies. The estimated meridional heat transport by eddy movement is similar in magnitude and spatial structure to previously published eddy covariance estimates from models, and the eddy heat and salt transports both are a sizeable fraction of their respective total transports.