Aims. We searched the first set of data released by the CoRoT mission for red stars showing long-period variability (LPV). These stars are likely highly evolved stars on the upper giant branch. The high time resolution of the CoRoT data allows us to study the light curves of these variables in unprecedented detail. Methods. Light curves of the reddest stars were extracted from the CoRoT public archive. The search for LPV was done by visual inspection. Instrumental artefacts were removed, and the final light curves were inspected for signs of variability on time scales of hours to a few days. Such abrupt and irregular light curve changes have been reported in the literature to occur in these stars, but their origin is not yet understood. Results. We detected 52 LPV candidates in four CoRoT fields, most of them previously unknown. No signs of any short-time light changes on time scales between 20 min and a few days and an amplitude exceeding a few mmag could be detected. We conclude that if these short-time variations are physical reality, they must be quite rare events, and we derive a 95% confidence upper limit of 0.15 events per star per year. Any deviations from a perfectly smooth long-time variation are occurring on time scales longer than approximately 10 days.