Would it be right to say that the transaction fees would also drop which could result in more movement of coins?
Transaction fees will not drop. The contrary: the competition will be much higher than today, and that's why it is considered enough of a reward. If demand for transaction confirmation is high, while supply remains the same (4 MB block size), then the price for block space skyrockets.
Ok, that makes sense, as above though I dont think it will come to that any time soon.