I think it has to do with inductance. E.g. B-field points from right to left; you move conducting loop left to right.
As you move it left to right, the magnetic flux through the loop is getting stronger; in other words, the change in magnetic flux is toward the left. Faraday's Law of Inductance (and I believe one of Maxwell's equations) states that the induced electromotive force and subsequent current will act OPPOSITE to the direction of the change in magnetic flux; in other words, d(magnetic flux) = -E (E is electromotive force, or induced voltage). So for the above example, if d(magnetic flux) points to the left, a current will be induced such that it will produce a magnetic force to the right (using the thumb rule for closed loop circuits, you'll see this induced current will be counterclockwise).