What is providing the force required to move the conductor in a changing magnetic field? Earlier, I came across this problem. It goes as,
"The current in the long wire is suddenly switched off. Neglecting gravity, what is the velocity acquired by the loop"
Attached figure:

Now, my doubt is, who is providing the force for the loop to move?
I've learnt that Magnetic fields don't do work. (But they can convert one form of energy to another).
In this case, however I thought of the following possibilities:
1) You need to do work in order to stop the current in the long wire, and that work is used to move the loop.
2) As there is a change in magnetic flux in the loop, a current is induced. Due to mutual inductance, A new current is induced in the wire. And hence, the loop gets moving. (Two current-carrying conductors show attraction/repulsion.)
(Ps: I personally think that the second possibility is more plausible. But here's another problem that I'm facing. Let's say that the second argument is true. Then that means that there is a change in flux through the loop due to this "New" current in the wire. So, the loop changes its current. So the wire changes its current too (by the same mutual inductance argument). And this keeps on going.)
(PPS: Any help is appreciated in trying to clear up my concept.)
 A: Your question is developed from the assumption,

I've learnt that Magnetic fields don't do work.

In comments you showed why the Lorentz force due to a magnetic field does no work, which is correct. And when we have a static magnetic field, the Lorentz force is the way it interacts with charged particles. So it's fair to say that static magnetic fields do no work. 
But changing magnetic fields produce (non-conservative) electric fields according to according to the Maxwell-Faraday equation
$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$
Or, in integral form,
$$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}  = - \operatorname{\frac{d}{dt}} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} $$
And it is common (even expected) this electric field to do work on charged particles that interact with it. 
In the case of a current through a loop of wire that is suddenly shut off, you're dealing with a changing magnetic field, not a static one.

You need to do work in order to stop the current in the long wire, and that work is used to move the loop.

To stop the current in the wire, you actually need to absorb energy from the magnetic field, changing it to some other form. You don't need to do work, you need to allow work to be done on you (or some other part of your system). 

As there is a change in magnetic flux in the loop, a current is induced. Due to mutual inductance, A new current is induced in the wire. And hence, the loop gets moving.

I think this is more or less the same argument I made above.
One quibble is when we talk about magnetic induction, we usually say an electric field or a electromotive force is induced, rather than saying a current is induced. 
A: 
who is providing the force for the loop to move?

If you're asking which force impacts momentum to the loop, it is mostly magnetic force from the wire acting on the current forming mobile charge carriers in the conductor of the loop. The current  in the loop is present due to induced electric field of the wire.
If you're asking which force does work on this loop to increase its kinetic energy, it is the force of the mobile charge carriers pushing on the rest of the conductor (this is not magnetic, and hence can and does work on the conductor). As a result, conductor gets moving and the current is decreased by this (due to motional EMF induced due to magnetic field of the wire).

I've learnt that Magnetic fields don't do work. 

This is an imprecise and misleading statement. There is a grain of truth in the sense magnetic force on charged particle does no work on this particle. But when we are talking about other magnetic forces, such as magnetic force on a conductor, this force does work on the conductor (it only does not work on the charged particles inside).
The thing about conversion of magnetic energy into matter energy is that there has to be some motion and some working happening, but the force that works is not the Lorentz force on charged particle, but it is either induced electric force, or internal force between the mobile charges and the rest of the conductor.

1) You need to do work in order to stop the current in the long wire, and that work is used to move the loop.

No, there has to be a force to stop the current, but in doing so, work is actually extracted. So experimenter does not need to supply work. In practice, electric resistance in the wire together with induced field of the loop will stop the current in the wire. Part of energy is extracted by the loop, other part transforms into heat in the wire.

2) As there is a change in magnetic flux in the loop, a current is induced. Due to mutual inductance, A new current is induced in the wire. And hence, the loop gets moving. (Two current-carrying conductors show attraction/repulsion.)

Well, due to mutual inductance the wire current is influenced. There is no new current, the one current that is in the wire changes differently in time due to action of the loop.
