Suppose you have a conducting circular wire loop and a magnet moving towards each other. They move along the $z$ direction with nonrelativistic constant speed $v$. Let the $B$ field of the magnet in its reference frame be parallel to the $z$ direction: $$\vec{B} = (0,0,-z),$$ so the strength of $\vec{B}$ linearly decreases along $z$. Let the surface normal of the area enclosed by the wire loop be pointing in the $z$ direction too. Now here is my problem: If I'd like to calculate the fields and forces acting upon charges in the wire, I get different results.
In the magnet's reference frame, there is a static magnetic field $\vec{B} = (0,0,-z)$, and the wire loop is moving with a constant velocity $\vec{v} = (0,0,1)$ towards the magnet. In this case there is no $E$ field, so the resulting force on charges is $\vec{F} = q\vec{v}\times\vec{B}$. Since $\vec{v}$ and $\vec{B}$ are parallel, the cross product is 0, so there is no force.
If I look at the wire's reference frame, there is a time-varying magnetic field producing an electric field. Here's where I don't quite get it. The electric field will be $$\vec{E}' = \vec{v}\times\vec{B}$$ according to Wikipedia (gamma is aproximately 1). In this case $$\begin{align} \vec{E}' &= 0 \\ \vec{v} &= 0 \\ \vec{F} &= q(\vec{E} + \vec{v} \times \vec{B}) = 0 \end{align}$$ just like in the other reference frame.
But if I try to use Faraday's equation instead, the wire loop sees a time-varying magnetic field, with a non-zero surface integral, so there has to be a non-conservative, non-zero $E$ field acting upon the charges.
I've tried this with different magnetic fields and every time my calculations fail miserably. I'm missing something somewhere, but I don't know why.