Question:

In that piece of a circuit outside of any batteries or power supplies what is it that actually pushes charged particles through the circuit?

Answer:

By definition a place of EMF that is an ideal battery or power supply maintains a constant electric potential difference between its terminals. Now electric potential is somewhat we use to characterize an electric field. So if there is an electric potential there have to be an electric field. Undoubtedly a seat of EMF creates an electric field in the circuit in the same direction as the current. Certainly the current is a result of this electric field as it is the electric field which exerts the force on the charged particles which makes them move around the circuit. One hardly ever hears of the electric field in a circuit because it is not useful in circuit calculations and it is difficult to measure.