Is ghusl obligatory upon the continuous emission of semen, or is it sufficient to perform ghusl once and change clothes, and does the emission of semen invalidate the fast?

The discharge mentioned by the questioner is most likely madhiy (pre-seminal fluid), vaginal discharge, or wady (a thick white fluid that comes out after urination). None of these necessitate ghusl (ritual bath) nor invalidate the fast. And even if it were semen, a wet dream not accompanied by the emission of semen does not necessitate ghusl, and similarly, semen that is emitted while awake without lust.