Under voltage can be just as bad as over voltage. When you supply less voltage to say an audio circuit or to a rf circuit it can cause troubles. Circuits are designed for a certain voltage and current. When you change the voltage the biasing of the transisters can be effected. As for the motors lower voltage will not effect them the same way. raiseing or lowering the voltage on a motor changes the speed lower voltage lower speed and vice versa. I am an electronics person by trade.
Let me give you an example:
I once had to repair an 8 track car tape player. It played fine till you turned the volume up or went to change tracks then it went screwy with the audio comming out of the speakers it developed a what we call motor boating. I changed solenoids and motors still had the fault. The I started checking the voltages on the battery side of the on off switch I had 12 volts on the other side of the switch 11.5 you wouldn't think that .5 volts would do much but it did. What had happened a tiny bit of carbon had formed on the contacts of the power switch and dropped .5 volts. That was enough to throw the bias voltage off on the output transistors causeing them to go into motorboating when you turned the volume up or changed tracks.
Increasing the current capacity of the batteries does not effect the system it just means the system has more available current it can use and under normal running it will run longer than a lower capacity. simple test take a 1.5 light bulb connect it to a AAA battery and record how long it lights then connect the same bulb to a D battery it will light for many times longer than the AAA battery.
Sorry for going on so long