The onset of variability in red giant stars is studied from the data collected by the `Long-Term Photometry of Variables' project which operates at ESO (La Silla) since 1982. The differential nature of the observations combined with a multi-night reduction algorithm enables to study the variability of over 50 red giants with an accuracy of the order of 2 to 3 millimagnitudes on time scales ranging from days to years. All red giants with spectral types late G to early K (i.e. b-y~0.6-0.8) are stable at the level Ï _y_<6millimag (where Ï _y_ is the standard deviation of the Stroemgren y magnitude). Red giants with later spectral types are all variable, and it is found that the minimum variability level Ï _y, min_ increases with increasing b-y, thus defining a minimum-variability boundary. The new data obtained in this paper reveal that the time scale of the associated variability increases with increasing b-y (and thus amplitude), and that the stability of this time scale improves concomitantly. More precisely, irregular variations on time scales of 5 to 10-days (and Ï _y_~7millimag) characterize red giants with b-y~1.0, whereas the variations (with Ï _y_~100millimag) become more regular with periods of the order of 50d for red giants with b-y~1.3. A radial-velocity jitter (of the order of 1.5km/s r.m.s.) is associated with this photometric variability, and suggests that stellar oscillations may be responsible for the observed variations.