We present spectra of 1142 colour-selected stars in the direction of the Sagittarius Dwarf Spheroidal (Sgr dSph) galaxy, of which 1058 were taken with VLT/FLAMES multi-object spectrograph and 84 were taken with the SAAO Radcliffe 1.9-m telescope grating spectrograph. Spectroscopic membership is confirmed (at >99 per cent confidence) for 592 stars on the basis of their radial velocity, and spectral types are given. Very slow rotation is marginally detected around the galaxy's major axis. We identify five S stars and 23 carbon stars, of which all but four carbon stars are newly determined and all but one (PQ Sgr) are likely Sgr dSph members. We examine the onset of carbon richness in this metal-poor galaxy in the context of stellar models. We compare the stellar death rate (one star per 1000–1700 yr) with the known planetary nebula dynamical ages and find that the bulk population produce the observed (carbon-rich) planetary nebulae. We compute average lifetimes of S and carbon stars as 60–250 and 130–500 kyr, compared to a total thermal-pulsing asymptotic giant branch lifetime of 530–1330 kyr. We conclude by discussing the return of carbon-rich material to the interstellar medium.