The F-19 production during the first dozen thermal pulses of AGE stars with masses M and metallicities Z (M = 3 M., Z = 0.02), (M = 6 M., Z = 0.02) and (M = 3 M., Z = 0.001) is investigated on grounds of detailed stellar models and of revised rates for N-15(alpha,gamma)F-19 and O-18(alpha, gamma)Ne-22. These calculations confirm an early expectation that F-19 is produced in AGE thermal pulses. They also enlarge substantially these previous results by showing that the variations of the level of F-19 production during the evolution is very sensitive to the maximum temperature reached at the base of the pulse. These variations are analyzed in detail, and are shown to result from a subtle balance between different nuclear effects (mainly F-19 production or destruction in a pulse, and N-15 synthesis during the interpulse), possibly superimposed on dilution effects in more or less extended pulse convective tongues. Our calculations, as most others, do not predict the third dredge-up self-consistently. When parametrized, it appears that our models of intermediate-mass AGE stars are able to account only for the lowest F-19 overabundances observed in solar-metallicity MS, S and C stars. That conclusion is expected to hold true for low-mass stars when fluorine production results from secondary C-13 only. Massive AGE stars, on the other hand, are not expected to build up large surface F abundances. Therefore, the large fluorine overabundance reported for the super Li-rich star WZ Cas (where hot bottom burning is supposed to be operating) remains unexplained so far. Our results for the (3 M., Z = 0.001) star indicate that fluorine surface overabundances can also be expected in low-metallicity stars provided that third dredge-ups occur after the early cool pulses. The relative increase in the surface F-19/C-12 ratio is, however, lower in the low-metallicity than in the solar-metallicity star. No observations are reported yet for these stars, and are urgently called for.