We consider the production of dynamic nuclear spin polarization (DNP) in a two-electron double quantum dot, in which the electronic levels are repeatedly swept through a singlet-triplet avoided crossing. Our analysis helps to elucidate the intriguing interplay between electron-nuclear hyperfine coupling, electronic spin-orbit coupling, and electron and nuclear Larmor precession in an externally- applied magnetic field, in guiding the production of DNP. In particular, we specifically address the roles of multiple nuclear spin species. By treating the nuclear spin dynamics semi-classically, we identify two contributions to the DNP production rate, a “geometric contribution” and a “dynamic contribution,” which depend in very different ways on control parameters such as the sweep rate and holding time near the level crossing. We find that the dynamical contribution dominates the DNP when the system is held near the singlet-triplet avoided crossing for a time on the order of the nuclear Larmor period. Detailed numerical calculations provide a physical picture for understanding the oscillations observed by Foletti et. al. in arXiv:0801.3613.