If someone told you that magnets don't feel a force from magnetic fields then you may have been misled. But if you want to use the Lorentz Force Law then consider using all wires or (more realistically) all resistors.
So you could have a super large parallel plate capacitor (make it planet sized) and have it charged up a lot and have two small thin resistors (with large resistance) near the center connecting the two plate. The resistors can run parallel to each other about a meter apart. And next to one of the long thin resistors you could have a series of batteries. So imagine a series of shorter resistors all connected in series with equally long wires and next to each battery you have a battery that isn't connected.
With the batteries disconnected you would have two resistors, each with a steady current, each producing a magnetic field and each feeling a force from the field of the other.
If you then hooked up all the batteries to the resistors it is next to by moving the battery to where the wire was then the current in that series of resistors changes right away and it starts feeling a stronger force right away.
But a meter away over at the other resistor, the field there is still due to the old current (new current doesn't change the magnetic field far away fight away) so it still feels a smaller force.
This is becasue wires don't exert forces on each other. Wires exchange momentum with the fields right where they are, and changes in the fields propagate at the speed of light and only when a new field value gets to a distant location does that distant location start exchanging momentum with the fields next to it at a different rate.
As for whether a wire moves up, it can only get momentum from the field and it only gets the opposite of what it can give. So wires move a certain way by giving an equal and opposite momentum to the fields where it is located.
Now, when the mobile charges feel a force they get deflected outwards from the wire, which creates a charge imblance in the wire that pulls it oppositely and pulls the nonmobile charges in the wire the direction the mobile charges were pulled. That's how the whole wire moved even though only the mobile charges felt a magnetic force. But that's also how work gets done, from the electric force.
So you do have limits to how much can be done. And the batteries are getting drained as well. If you track the energy and the momentum flow you'll see them coming in through the sides of the resistors.
I don't see a clear explanation why the force exerted on the electrons in the wire could not be transmitted to the B field source
A wire feeling a force from a magnetic field exchanges momentum with the field. A wire getting energy (which doesn't happen from magnetic forces) takes energy from the field. So energy and momentum went into the field long ago. Energy and momentum moved through space in the fields. Energy and momentum gets exchanged with the wire based in the current and charge in the wire and the fields at the wire.
The same arguments could be made for electric forces. The energy and momentum went into the field long ago, the energy and momentum moved around in space through the fields. And the charges get energy and momentum from the fields where they are located by exchanging energy and momentum with the fields right there.