  Variability is examined in over 2.6 million stars covering 11 square degrees
of the core of the Sagittarius dwarf spheroidal galaxy (Sgr dSph) from VISTA
Z-band observations. Generally, pulsation on the Sgr dSph giant branches
appears to be excited by the internal kappa mechanism. Pulsation amplitudes
appear identical between red and asymptotic (RGB/AGB) giant stars, and between
unreddened carbon and oxygen-rich stars at the same luminosity. The lack of
correlation between infrared excess and variability among oxygen-rich stars
indicates pulsations do not contribute significantly to wind driving in
oxygen-rich stars in the Sgr dSph, though the low amplitudes of these stars
mean this may not apply elsewhere. The dust-enshrouded carbon stars have the
highest amplitudes of the stars we observe. Only in these stars does an
external kappa-mechanism-driven pulsation seem likely, caused by variations in
their more-opaque carbon-rich molecules or dust. This may allow pulsation
driving of winds to be effective in carbon stars. Variability can be simplified
to a power law (amplitude proportional to L/T^2), as in other systems. In
total, we identify 3026 variable stars (with r.m.s. variability of delta Z >~
0.015 magnitudes), of which 176 are long-period variables associable with the
upper giant branches of the Sgr dSph. We also identify 324 candidate RR Lyrae
variables in the the Sgr dSph and 340 in the outer Galactic Bulge.
