Faraday's law states that as the magnetic flux through a loop of wire changes, an EMF is induced around the wire that is proportional to the rate at which the total flux through the loop is changing…
But of course the wire cannot directly measure the flux inside the loop. The electrons in the wire can only be affected by fields in the wire. There must, therefore, be an alternative formulation of Faraday's law that integrates around the loop instead of across the surface that it bounds.
This is easy to imagine if field is constant and the loop is moving. From the motion of the wire and the flux vector at every point on the wire, you can determine how much flux is entering or leaving the loop at every point and sum them up to get total $d\phi_B/dt$. In fact the EMF will be the sum of the Lorentz forces.
But how does this work if the loop is stationary and the flux is changing arbitrarily? We can imagine divergenceless fields that pass through the loop, but do not intersect the loop at all. If such fields were to wax and wane in place, then no integration around the loop could possibly evaluate the changing flux.
So… such divergenceless fields must not be possible. There must be some further constraint on the structure of magnetic fields that lets the flux change be calculated with a line integral. It must be possible to derive the sum of the amount of flux entering of leaving at every point from the flux vector and its derivative alone around the loop.
What, exactly, is that constraint on the structure of magnetic fields, and what is the line integral formulation of Faraday's law that it enables?
Progress:
Some answers below mention this intuitive picture of "field lines" that we draw for ourselves, and assert that these lines cross the loop of wire as the field changes.
That implies that the field lines move.
Motion, though, is not really a property of the field, so why is it a property of these imaginary lines that we draw?
I've certainly seen animations of a waxing dipole field, with field lines moving out from the center into space as the field strength increases. The line density indicates flux density, and those lines definitely would cross a loop of wire as those answers below suggest.
I haven't yet figured out if the number of imaginary line crossings is a linear function of the local properties of the field, but I have figured out why that animation makes sense for dipole fields.
Given a dipole field at the origin, so:
$$
\vec B(\vec m, \vec x) = \frac{3\vec x(\vec m \cdot \vec x)}{|x|^5} – \frac{\vec m}{|x|^3}
$$
… where $\vec m$ is proportional to the dipole moment, we find that
$$
\vec B(\vec m, \vec x/\alpha) = \alpha^{3}\vec B(\vec m, \vec x)
$$
i.e., a bigger dipole field is the same as a stronger dipole field.
So imagine that we made a 3D model of a dipole field with many field lines, at a density proportional to flux density. We put an imaginary loop near this model, and then we gradually scale the model up by 2.
As we scale this model, field lines will cross the loop. By the equation above, if we had scaled a real dipole field up by the same amount as our model, there would be 8 times as much flux passing through the loop. The scaling, though, has moved our modeled flux lines farther apart, reducing their overall density by a factor of 4, so there are only 2 times as many modeled lines passing through the loop.
In the net, as we scale our model up by a factor of $\alpha$, the result is an accurate model of multiplying the field strength by a factor of $\alpha$, with lines crossing any loop at a rate that sums up proportional to $d\phi_B/dt$.
Since real magnetic fields (and maybe all divergenceless fields?) are summations of dipole fields, we can accurately model a field using field lines that expand outward from, and collapse inward to, the dipole centers. Since the dipole centers for the fields around magnets and coils are all inside the magnets and coils, this makes sense intuitively and visually.
For the kinds of strange fields I was wondering about above, that could be zero everywhere along a loop of wire, for example, the corresponding dipole centers would be spread out through space, and imagining lines that emanate from them or collapse into them would be very difficult.
Resolution:
I see now that the magnetic field doesn't need to have any special form, other than being divergenceless, because the EMF around a closed loop just isn't determined by the local properties of the magnetic field.
The line integral form of Faraday's law (for a non-moving circuit) is just:
$$
\mathcal E=\oint\frac{\partial \vec A}{\partial t}\cdot\mathrm d\vec l
$$
where $\vec A$ is the magnetic vector potential, of which $\vec B$ is the curl. This can just not be expressed in terms of local properties of B, i.e., without integrating over space.
I was fooled by the idea that the "field lines" we imagine are representative of the magnetic field only, and that they should move according to changes in their own local properties. But it turns out that if you want to animate the field lines in order to represent a changing field in such a way that the number of lines crossing loops reflects Faraday's law, then the velocity of those lines needs to be
$$
\frac{\partial \vec A}{\partial t} \times \frac{\vec B}{|\vec B|^2}
$$
(and this can probably be simplified using some vector calculus identities that I don't know).
The calculation of $\vec A$, and these velocities, requires an integration over space of some sort. In retrospect, I suppose this is not surprising — since each line represents a constant amount of flux, then the position of the nth line will be determined by an integration of flux over space, no matter where you "start" from.
Thanks to all those who helped.
Best Answer
You don’t need an alternative formulation of Faraday’s law, just the inclusion of Gauss’ law for magnetism as well. That law implies that you can describe the magnetic field in terms of magnetic field lines that form continuous loops. It is that property which resolves the issue you consider.
Specifically, the only way for the flux through the inside of the loop to change is for a magnetic field line to cross the wire. If we try to increase the flux by avoiding the wire then we inevitably have a line that goes through the inside of the loop twice, once in each direction. Such a line contributes no net flux, it cancels itself out.
The only way to change the amount of flux inside the loop is for a field line to cross the wire. It is that crossing of the wire that induces the EMF. As it crosses the wire, there is no need to imagine any remote influence. What affects the EMF occurs only at the wire itself.