To understand what happens during a transition, you'll need to go back to the basics.
The negative conductor initially has a surplus of electrons. These electrons don't like to stay close to each other and given enough time they will adopt a configuration which minimizes the potential energy as much as possible. In this case, that configuration is where the charge spreads out on the surface of the conductor. It would be even better if the electrons left the conductor but unfortunately, the poor electrons are bound to the conductor.
When you bring the negative conductor in contact with the neutral conductor, the charges on the surface near the point contact immediately get pushed into the neutral conductor. As of now, the electrons aren't aren't aware of the fact that the neutral conductor is grounded. The electrons keep moving into the neutral conductor. As the electrons near the point of contact move into the neutral conductor, the electrons near the near the point of contact see that the electron density is decreasing, hence, they start moving closer to the point of contact and then ultimately to the neutral conductor. At one point when the electron density reaches a certain point, the electrons in the conductor which was once neutral face repulsive forces and this push them into the wire and they start flowing into the ground.
As time passes, all the charge would have left both the conductors as the earth can be considered to be a big sink. This would leave both the conductors neutral and with zero potential.