Exactly. Eddy currents are induced in metal plates by Faraday's Law, usually dB/dt. Also, as you point out, eddy currents can be generated in a metal plate moving in a non-uniform magnetic field. There are many many swirls, as you also point out. The eddy currents not only reduce the magnitude of B (per Lenz's law), but also create heating. The heating is often the more important problem, like in transformer laminations. This is why transformer laminations not only have grain-oriented steel to improve the magnetic properties (including lower hysteresis losses) but also have higher resistivity to reduce the eddy currents. Eddy currents can be generated in any conductor, including non-magnetic materials, like aluminum and copper (including copper wires).
Bob S