A changing electric field produces magnetic field and vice versa. Does that mean that this process will carry on forever? Think of a circuit with a capacitor. The magnetic field due to the current at a point on the wire (by the Ampere-Maxwell law). But the current was changing with time, so it also meant that the magnetic field changed. And a changing magnetic field produces an electric field, so we have to go back again from the start by Ampere's law. It seems that this will go on forever. What is the final magnetic and electric field that I have to calculate?
The idea in the comments above is a good one. The reason you don't need to worry about the order is that you're looking for an equilibrium solution. In terms of going on forever, it's broadly true. I mean electro magnetic radiation is exactly the kind of effect you're talking about. In a circuit there is normally a dissipative term, but in a steady state with a power supply you just need to solve the equations simultaneously