A comprehensive integrated model of stochastic convective transport in a solidifying binary melt is presented in this work. The detailed transport phenomena in the particle and bulk phases are coupled together through a stochastic Eulerian-Lagrangian formalism, capturing the physical mechanisms and consequences of particle agglomeration, de-agglomeration, and the underlying hydrodynamic interactions. The interactions between random thermo-fluidic fluctuations in the continuum carrier phase and the associated growth/dissolution of particle phases are modeled by employing a Langevin formalism. Representative case studies highlighting the implications of the dynamics of the fragmented dendrites on the overall convective patterns in an electromagnetically-stirred semi-solid binary melt are subsequently presented, so as to emphasize on the comprehensive physical bases and to illustrate the fundamental approach of the present predictive methodology.