A positive charge moves towards a wire with positive current. How are the effects of the magnetic field explained upon a Lorentz transformation? Consider a positive charge $P$ moving towards a wire that has the same density of positive charges and negative charges, but so that the positive charges are moving to the right.  Then the magnetic field should cause $P$ to move to the left.
Now consider the Lorentz transformation that makes $P$ still.  Now the wire is moving to approach $P$.  But because $P$ is still, it cannot be affected by any magnetic effects, so any force on it is caused by electric effects.
So now we have a wire with positive charges moving to the right approaching $P$, and somehow this creates an electric force on $P$ to move to the left.  How?
 A: 
because P is still, it cannot be affected by any magnetic effects

This assumes that the magnetic field at P is static. A changing magnetic field can cause even a stationary charged particle to move.
Explanation Using Magnetic Fields
In the frame of reference in which P is still, the wire moves toward the particle, causing the wire's magnetic field at the particle to change (strengthen in this case), which creates an electromotive force via electromagnetic induction that moves the particle parallel to the wire even though the particle had been stationary. This is the effect that an electrical generator makes use of - a magnet rotates in a coil of wire and the changing magnetic field induces a current in the wire.
Explanation Using Electric Fields and Relativity
We can also do away with magnetic fields and explain this effect purely through electric fields and special relativity: In the reference frame of the charged particle, we can note that the positive charges are moving diagonally down and to the right, causing them and their electric fields to be Lorentz contracted in that same direction. As you can see in the image below, this means that the electric forces repelling the particle from the right are stronger than those repelling it from the left, resulting in a net force that pushes it to the left. Note that since the negative charges are just moving straight down, they exert no net horizontal force on the particle.

In terms of net vertical force, since the vertical components of the velocities of the positive charges are equal to those of the negative charges, the same is true of their contraction and thus of their E fields. This combined with the fact that the positive and negative charges are equally numerous means that the vertical components of their E fields cancel out, leaving the force to the left as the only net force on the particle.
See the "A Charge Moving Perpendicular to a Wire" section of "Magnetism, Radiation, and Relativity" by Daniel Schroeder (of which there's a shorter version and a longer version), which summarizes the more detailed treatment in Electricity and Magnetism, 3rd Edition (pp. 265-267) by Edward Purcell and David Morin.
