ABSTRACT Fast radio bursts (FRBs) are transient intense radio pulses with duration of milliseconds. Although, the first FRB was detected more than a decade ago, the progenitors of these energetic events are not yet known. The currently preferred formation channel involves the formation of a neutron star (NS)/magnetar. While these objects are often the end product of the core-collapse (CC) explosion of massive stars, they could also be the outcome of the merging of two massive white dwarfs. In the merger scenario the ejected material interacts with a constant-density circumbinary medium and creates supersonic shocks. We found that when a radio pulse passes through these shocks the dispersion measure (DM) increases with time during the free expansion phase. The rotation measure (RM) displays a similar trend if the power-law index, n, of the outer part of the ejecta is >6. For n = 6, the RM remains constant during this phase. Later, when the ejecta move into the Sedov–Taylor phase while the DM still increases, however, with a different rate, the RM reduces. This behaviour is somewhat similar to that of FRB 121102 for which a marginal increase of DM and a 10 per cent decrease of RM have been observed over time. These features are in contrast to the CC scenario, where the DM and RM contributions to the radio signal always diminish with time.