Context: Solar eruptions, such as coronal mass ejections (CMEs), are often accompanied by accelerated electrons that can in turn emit radiation at radio wavelengths. This radiation is observed as solar radio bursts. The main types of bursts associated with CMEs are type II and type IV bursts that can sometimes show movement in the direction of the CME expansion, either radially or laterally. However, the propagation of radio bursts with respect to CMEs has only been studied for individual events. Aims: Here, we perform a statistical study of 64 moving bursts with the aim to determine how often CMEs are accompanied by moving radio bursts. This is done in order to ascertain the usefulness of using radio images in estimating the early CME expansion. Methods: Using radio imaging from the Nac{c}ay Radioheliograph (NRH), we constructed a list of moving radio bursts, defined as bursts that move across the plane of sky at a single frequency. We define their association with CMEs and the properties of associated CMEs using white-light coronagraph observations. We also determine their connection to classical type II and type IV radio burst categorisation. Results: We find that just over a quarter of type II and half of type IV bursts that occurred during the NRH observing windows in Solar Cycle 24 are accompanied by moving radio emission. All but one of the moving radio bursts are associated with white--light CMEs and the majority of moving bursts (90%) are associated with wide CMEs (>60 degrees in width). In particular, all but one of the moving bursts corresponding to type IIs are associated with wide CMEs; however, and unexpectedly, the majority of type II moving bursts are associated with slow white-light CMEs (<500 km/s). On the other hand, the majority of moving type IV bursts are associated with fast CMEs (>500 km/s).
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