The first simultaneous measurements of discrete chorus emissions on multiple spacecraft, realized in the context of the Cluster mission, revealed a rather unexpected frequency difference of around 1 kHz between nearly identical discrete elements observed on different spacecraft <Gurnett et al., 2001>. This frequency difference is interpreted herein as a natural outcome of the dependence of the whistler-mode refractive index on the wave normal angle between the wave vector k and the static magnetic field B0 and the rapid motion of highly localized source region(s) of chorus of 400 km to 1700 km in extent along the field line, but only less than 100 km transverse to the magnetic field, and moving at speeds of 20,000 km/s to 25,000 km/s. Wave packets emanating from the localized regions propagate to two spacecraft at different wave normal angles, and are observed at different frequencies due to the differential Doppler shift between the two spacecraft. These differences in frequency, as well as the different times of arrival of the similar emissions at the different spacecraft, provide a unique opportunity to estimate the source characteristics, using a model involving rapidly moving sources traveling at speeds comparable to the parallel resonant velocity of counterstreaming electrons moving along the Earth's magnetic field lines. We report the determination of chorus emission source region motion for two different cases observed during 2000--2001, where these differences in frequency were readily observable due to the relatively large separation of the Cluster spacecraft. We also report a case in 2002 where the spacecraft separations were smaller, so that these frequency differences were not as evident but nevertheless measurable. In general, our results provide the first experimental evidence that the sources that generate the discrete chorus emissions are in rapid motion.