According to page-1262 of Fundamentals of Physics by Halliday, Resnick & Walker (10th edition),
Both the electrons in the conduction band and the holes in the valence band serve as charge carriers. The holes do so by permitting a certain freedom of movement to the electrons remaining in the valence band, electrons that, in the absence of holes, would be gridlocked.
So, we can understand that the holes move by means of the "gridlocked" electrons. My guess is, free electrons don't combine with holes as they (free electrons) are in a higher energy state than the "gridlocked" electrons. Consider this, if free electrons had a tendency to combine with holes, why were the free electron-free hole pair created in the first place? The free electron and hole would've combined and would've ceased to exist the moment they were created. Is my logic sound?
So, do free electrons, at any point in transistor physics, combine with holes to become neutral?