I know that a battery does work when it takes positive charge from one plate of capacitor and deposits it to the other plate, thereby creating equal and oppositely charged plates having electric field in between them.However I am unable to understand how this process works.

Let us consider a battery with EMF E connected to an uncharged capacitor and a resistance R. When the circuit is closed, an initial current equal I=E/R is established in the circuit which drops exponentially. In a small time interval dt, a positive charge dq=I*dt will be deposited on one plate of capacitor, which will attract same amount of negative charge on the other plate and hence establish an electric field between them. The electric field of battery doesn't do any work initially since the capacitor is uncharged in the beginning. I believe that later if battery adds more charge to the already present charge, it will have to apply force against the electric field of already deposited charges and thus do work in the process. Is my assumption correct?

However if I place a mechanical source of energy between plates of an uncharged capacitor such as a conveyor belt (instead of a chemical one which is a battery) which takes a positive charge dq from one plate and deposits it on the other plate, it has to do work against the electric field between positive and negative charge of magnitude dq to create a charge separation.

So how come a battery doesn't do any work to create an initial charge separation of dq while a mechanical source of energy does? Are the two processes of charging capacitors different or am I missing something in the battery scenario?