Just for clarity, if I say positive y (direction) then I mean up, and if I say positive x (direction) I mean to the right.
We will follow the life of an electron, we can call it El.
So, our conduction electron El is happily moving on the top wire, to the right. It's guided by electrostatic forces in the sense that when it bumps into something that slows it down, the conductions electrons in front keep going (so postiive charge imbalance there, attracting it) and the electrons behind it keep building up (negative charge imbalance there, pushing it forward). Same if something pushes it extra. It's like traffic, there are consequences for stepping out of line, in this case electrostatic consequences.
OK, so this electron El will go into the wire on the right for the same reason, electrostatically the electrons in front of El went that way, so there is an opening (will be a charge imbalance), but turning that corner all those protons are moving at speed v to the right. So even though there isn't an earlier electron to "follow" there is a charge imbalance there, so electrostatically it gets pulled that way, and will continue until it also is moving at speed v to the right. But now something new is happening.
Now El has a velocity to the right, and so can experience an magnetic force downwards, and the electrons in front there are experiencing the same kinds of forces, so everyone can have some force per unit charge in the direction of the circuit element. We can measure this, and it's called EMF. So the conduction electron gets an EMF because of the rightwards component of the velocity.
This is not work because force is orthogonal to the motion (a more precise analysis would note that the average velocity of the particle needs to get it from the top at one time to a point below it ... and to the right since by the time it gets to the bottom the wire is farther to the right, but there will still be no work in this more careful analysis, and it's only the part of v orthogonal to the circuit element and the magnetic field that contributes to the EMF, so it's not an entire fraud. An even more careful analysis would have a sea of random velocities with a net bias for a drift velocity, but again it's the average velocity that contributes a net effect, and again only the rightwards component of the velocity that matters to the EMF).
Now the downwards motion of the electrons also contributes to a magnetic force. And there is the normal electromagnetic pull and push downwards (traffic opening up in front and honking behind) and the magnetic force helping now too (because of the rightwards velocity). This has a magnetic effect to send the electrons onthe left of the wire out (farther left) and those on the right deeper into the wire), and either way electrostatic forces respond to that charge imbalance by pushing in the opposite direction to keep the charge from building up an imbalance on the right (magnetism pushes charges on the right into the wire, but then electrostatic push back to the right, so the right side is a tiny bit charged, exactly enough to counter the magnetic force) and the charge imbalance on the left (magnetism pushes charges on the left out of the wire, but then electrostatic forces push back to the right, so the left side is a tiny bit charged, exactly enough to counter the magnetic force).
Basically, you have neutral wire and a pretty steady current (not changing fast relative to the speed of light) so magnetic forces do push the charges around, but always orthogonal to the velocity of any fixed charge and small charge imbalances counteract any deviation from the steady flow of charge.
Now, there is a current, and there is energy lost due to the current. Where does that energy come from? When the electrostatic forces smooth out the charges, the charges exert equal and opposite forces on those protons. So when you pull an electron on the right edge to the right, the protons get pulled left. And when you pull an electron on the left edge of the wire to the right, the protons get pulled to the left. The net effect is that either the bar is going to get pulled to the left, or else something else also needs to pull the bar to the right to keep it going at a steady speed.
So to summarize. The bar moving to the right means there will be increased flow in the direction of the circuit element, so we get an EMF. This flow is smoothed out by electrostatic forces so that it doesn't build up or get away from the wire. These electrostatic forces on the conduction electrons individually have equal and opposite forces, so the net total electrostatic force on all the conduction electrons is equal and opposite to the net electrostatic force on the other charges (the protons, bound electrons, etc.), and this either slows that moving rod down, or is opposed by a totally new force, the force pulling that rod over to the right.