I can't figure out your question, and wonder if it contains a typo.
However, consider a very simple system (simple in the sense that it's made of material you should know, and operates according to principles you've seen before). First use a conducting rod pointing in the x direction, and being moved at a steady speed in the y direction in a region with a uniform magnetic field in the z direction.
The magnetic force is our nonelectrostatic force. Due to the motion of the rod in the y direction the protons feel a force, but this just stresses the lattice and it strains the rod and it gets a bit bigger until there is an equal but opposite stress from the tension, so we can ignore the protons. Due to the motion of the rod in the y direction the conduction electrons feel a magnetic force so the charges pile up on one end, and leave an opposite imbalance on the other end. Eventually they pile up enough charge imbalance (like a capacitor) that the electrostatic force balances the nonelectrostatic force (the magnetic force in this case). During the time the rod charged its ends equally and oppositely the energy during the charging up came from the person pulling the rod. This is like a battery that is not hooked up to a circuit.
To hook it up to a circuit we can place a resistor parallel to the conducting rod but in in a region without a magnetic field (otherwise it'll be like another battery instead of like a resistor). We can then connect the rod and the resistor with two conducting rails that point in the y direction, but then the magnetic force just pushes them to one side of the rails, but doesn't drive them to one end or the other, so doesn't do anything important (the work function could be different on one side than the other). Now the charges won't pile up at the two ends of the rod, because each electron can just move away from the others down the rail, again there will be a very short period of time when the charges spread themselves out. But after they've done that, when the charge density isn't changing any more there are still electrostatic fields (the electric field caused by the location of charge density that isn't changing). And electrostatic fields have a curl of zero. So the forces are the electrostatic force which contributes no emf and the magnetic force which contributes only in the conducting rod.
So when the rod was by itself (no conducting rails, no resistor) it charged up its ends until the electrostatic force balanced the magnetic (nonelectrostatic) force. Later, when hooked up to circuit the electrostatic force smoothed the charge and current out to avoid charge build ups (unless there is a capacitor in the circuit), and so it wasn't fighting the magnetic force so the magnetic force can apply a force on the charges, the hand pulling the conducting rod can provide the energy and current can flow to the conductor where energy is lost as heat.
Other batteries work the same way, except they generate their charge buildup by different means (chemical, pressure, temperature gradients, etc.)