Let's establish a few facts and findings.
Magnetic shielding is different from blocking electric fields. in a Faraday cage, the electric field causes electrons to move towards one side, making an internal field canceling out the original. Magnetic shielding, on the other hand, provides a way through the shield and out, making sure the field does not pass through the prohibited area.
This paper has this picture, it's a great visual aid to understand this:
Note this is for visual guide, the actual study does not refer exactly to what I'm saying. So to sum up, you need a highly magnetic material to avert the field "strength" lines. Also note that the lower the frequency, the harder it get's to block them.
Now to address what happens when you push through. if the loop is completely inside the area of constant field, the force at the opposite ends are towards the same direction, look at this loop for example:
So electrons will accumulate on the side, opposite to the direction of the forces F1 and F2. that means you have positive charge at one end, and a negative at the other end, so no current is flowing in equilibrium, that is when the charges stop moving (can you see how this resembles a "source" of sort? imagine if you have something out of the coil and field system, through which electrons can travel, the beginning of "generating" electricity...). So now one might ask the question, is the wire loop already inside the field when it starts moving, or does it enter the area gradually?
Let's assume that it enters it gradually, Then you will have a current running through your coil while some part of your device is outside of the area with fields. You can deduce how it will look like if it starts moving in a field... (hint: short times and high amplitudes)
So if it is only the detection of current that you are looking for, both cases 1 and 3 will have current, but for a time. if it is anything else your looking for or specific values, that's a different story.
EDIT: just thought of a possible question, one might ask how could there be no current if the wire is moving? assuming the wire is neutral originally, ofc. If you are familiar with Faraday's expression of induction, you probably has seen it's mathematical form, that is:
$$emf = -N\frac{d\phi}{dt}$$
$$\phi = BA (Simplified)$$
Here, emf is the induced electromotive force, what makes the charges move, ϕ is the "amount" of magnetic field that goes through an area, and B is the symbol for magnetic field. if you are not familiar with differentiation, the fraction in the first equation just means rate of change of ϕ. The electrons do indeed have a speed, so they are having magnetic "effects", but they are not moving to the eyes of an observer travelling with the wire.
So for the ϕ to be variable with time, thus making an induced potential difference, either area or the value B (which is the magnetic field) must be changing. if the loop is moving in constant field and not changing it's shape, the area and the field values are constant, meaning no induced voltage.
