When crystals are grown from solutions, the driving force of the process is supersaturation.
When crystallizing from a melt, it is important to take into account the processes that affect the composition of the melt (thermal dissociation, evaporation, interaction of the melt with the environment), processes at the crystallization front, heat transfer processes in a crystal and melt, processes mass transfer (transfer of impurities due to convection and diffusion in the melt.
Agree with your information / explanation, and endorse it. In VDS-process, the entire detached crystals (L~ 65mm, d~12mm) have been grown in the vacuum sealed ampoules with identical growth conditions and parameter. The source materials (In, Ga, Sb) were taken in stoichiometric proportion. The increasing composition % (In or Ga) changed the mobility and conductivity; it reveals the changed properties along the growth direction.