Good curiosity! Sometimes people don't think about how the electricity travels as long as their circuit works but most Professional Engineers or hobiest would know this for sure.
Electricity flows with Voltage (energy of electrons or charge as charge is directly proportional to #of electrons). The copper is just a medium that allow electrons to move freely. It is the battery that is the source of electrons and holes otherwise battery would not have been called a source! So, the chemical reaction inside a battery generate cations and anions that are generated at two terminals "+" and "-". As long as they are connected through your circuit electrons and holes find a path to travel. An electron from a battery terminal doesn't necessarily have to flow from "-" end to another end of the "+
" terminal but think of them as like a series of electrons in a line (the motion is quite shattered inside bad conductors but pretty smooth and conductive in copper wire) push each other. Same thing with holes but in reverse direction, as absence of electron is hole.
So, depending on the voltage (which depends on intensity of chemical reaction inside a battery) and the circuit impedance, number of electrons/holes travel through copper. That electron might be belong to the copper or the battery as they tend to push each other to create a electron motion. Remember, electron travel speed is much slower (as they just have to hit and pass the energy to next electron) than the motion created by the real electricity.
After a long consumption of charge, the chemical process inside a battery slows down and therefore, the energy in your charge is less so, less force to push electrons and thus your battery starts giving lower voltage than its regular voltage.