The simple answer is that although it is hard to track
the motion of a single electron, the fundamental
characteristic of a circuit is that electrons in any
point in the circuit acquire a drift current, based
on the accumulation of charges in the circuit due to material
properties and driving effects (e.g. the battery).
In a series circuit with minimal inductance, this drift
current rapidly becomes constant across the entire circuit.
By "drift current," I mean the average charge flux (or e.g. in
the case of a capacitor, displacement current) through the
cross-section of the wire or circuit element we are considering.
For example, many electrons might be passing left or right
through the cross section, but if more electrons are going right
than left, then we say the drift current is to the left.
(Noting that electrons are negatively charged, of course.)
It is this average charge flux—this drift current—that
equals the current $I$ through the circuit. So, to conclude
the "answer" part of this response: even after the last
resistor, electrons will continue to move toward the
battery terminal, on average.
(To be repetitive), this is because in steady-state for a DC-powered series circuit, electrons drift from the negative terminal toward the positive terminal no matter where they are in the series circuit (excluding the battery, which pulls negative charges from the positive terminal and puts it back on the negative terminal, expending its stored energy in the process).
(Note: I have some other thoughts on this, including a rough explanation thinking of resistance as being due to an effective drag force, but I will need some time to collect my thoughts here. I'll probably edit this answer later.)