If we move the charge $+Q$ a little bit to the right or to the left, more negative charges will gather in the same direction, while positive charges on the external surface of the conductor won't move at all.
Why is that possible? Do the positive charges repel from each other more than how much they feel the electrons? In electrostatic conditions, the internal electric field is zero, but moving the charge $+Q$, why can't something happen? Maybe the internal field is zero, if the field from more negative charges on a side is counterbalanced by a field from more positive charges on the same side.
Edit: Thanks @Hari for the answer; what I don't get is why this happens:
Any region outside the cavity is unaffected by the movement of charges inside the cavity.
Ok, the internal electric field is zero, but this is true if and only if the molecules (and charges) of the conductor rearrange in order to generate an opposite and equal electric field to the "external" one (the one generated by the charge inside the cavity, external meaning not generated by the charges of the conductor themselves).
Does there exist a diagram with the field lines for this situation?
but with the opposite contributions that make zero the elctric field inside the conductor.
In other words, why the second situation cannot hold?
Here the answer is given in terms of work (though that sounds a bit strange to me, talking about the work done to move charges from infinity [??] to the side of the conductor), but what about the electric fields?