Let me go step-by-step.
First I will outline a simple transformer. There are two circuits next to each other (not touching). One is an electromagnet set-up (with AC supply) and the other has a galvanometer. Turning on the electromagnet induces an EMF and a current in the circuit next to it. Increasing the number of turns in the electromagnet increases the strength of the magnetic field and the induced EMF in the second circuit falls while its current rises. This I've been told is due to the fact that $P$ remains constant and $P=VI$.
Now I'd like to outline another setup. There is a coil in a circuit with no power supply. There is a galvanometer. I take a magnet and continuously thrust it into and out of the coil, inducing an alternating EMF and current. The speed with which I thrust the magnet is constant, and I believe that I do a certain amount of work by thrusting it. Therefore I have a constant power. Now I take a stronger magnet (so as to increase the strength of the magnetic field) and I repeat the experiment with the same speed of thrust (therefore same power). Both the induced EMF and the induced current increase.
Now correct me if I'm wrong. A moving magnetic field and a changing magnetic field both have the same effect for inducing EMF and current. My power input is constant in both setups. I increase the magnetic field strength in both setups. Both setups are virtually the same. The EMF falls while current increases in the first setup. Both the EMF and the current increase in the second setup. Why this difference in the results?