The electric current in a conductor is proportional to a quantity called the drift velocity, which is the average velocity of the electrons in the conductor.

In a conductor without an applied electric field (for example, a piece of wire with no battery connected to it), the drift velocity is zero. This is because the electrons are, on average, not moving in any particular direction. [Note that the electrons are still moving, and at very high speeds, but they rebound off of the atoms in the conductor at random angles which average out to no motion.] Upon application of a voltage, there is then a push toward motion in one direction through the conductor, a non-zero drift velocity proportional to the applied voltage is attained, and current flows in the conductor.

Since the current is a result of a net electric field in a particular direction, speeding or slowing the current can be achieved by increasing or decreasing the magnitude of that field. Electric fields follow the principle of superposition, so increasing or decreasing the field requires creating another electric field in the same location. As hinted in the question, fluctuating magnetic fields induce electric fields in conductors (typically called an electromotive force, EMF, in this context). This is Faraday's Law.

Magnetic fields also follow superposition, so 7 individual fluctuating magnetic fields around the circuit results in a single net fluctuating magnetic field, and a single net EMF. The direction of the EMF depends on the change in the flux and follows Lenz's Law.

Whether the EMF causes the current to flow faster or slower will thus depend on the direction of the magnetic field and the direction of change of that field (so it will reverse at times with the fluctuation). Realize that a field other than the induced EMF does not need to be present in the circuit. If this is the case, then the rate of current flow without the magnetic fields is zero, and faster or slower will depend solely on the induced EMF.

Note that the principles in this question are essentially those governing operation of DC generators and motors, except that the question has the magnetic fields fluctuating rather than the conductor moving ("fluctuating") through fixed magnetic fields.

Here is a video which clearly explains how these generators and motors work. (The video is old and long, but it is still one of the best I have found. A shorter one can be found here. )