thanks for the answers! The lecture notes were also a bit enlightening. So what I've gathered from this is that the electrons should be thought of as bound to the fluid/plasma, but being able to freely move inside it. I was worried about the changes in momentum by the Lorentz forces on ions and electrons cancelling out, but now I see that the surviving part, which pushes on the fluid as a whole, comes from the motion of electrons *within/relative to* the (electrically neutral) fluid.
Still, I am wondering if there are scenarios where a separation of charges *does* become important - i.e. the charges get ripped apart faster than they can recombine? Maybe at very high velocities/magnetic fields? Although I guess one would have to take relativistic effects into account then...