A curvatureless three-dimensional boundary-integral algorithm has been developed for tangential Marangoni stresses and used to study breakup and capture of two deformable drops for arbitrary drop-to-medium viscosity and thermal conductivity ratios in parallel and antiparallel arrangements of gravity and an applied vertical temperature gradient. When the driving forces are opposed, the previously observed inhibition of breakup by a weak thermocapillary effect for drops with equal viscosity and thermal conductivity ratios is shown to be almost exclusively the result of changing interfacial tension, with Marangoni stresses having virtually no influence. Alignment of gravity and the temperature gradient in the same direction enhances breakup more than opposing driving forces reduce it, with the limitation that the drops are moving toward a region of zero interfacial tension. The thermal conductivity ratio has negligible impact on these interactions. For bubbles, the effect of a temperature gradient on gravitational results is much less pronounced than for drops. Under certain conditions, parallel orientation of the driving forces weakly inhibits the capture interaction for bubbles, while antiparallel orientation enhances the phenomenon.