Check out this youtube video titled Einstein De Haas effect, uploaded by the University of Michigan Demo lab
The demo shows a torsion pendulum.
The amplitude of the swing is back and forth around a vertical axis. The amplitude of the swing increases because the swing is pumped.
The current in the surrounding coil is reversed in resonance with the natural frequency of the torsion pendulum. The Einstein De Haas effect is very small, the resonance setup accumulates the effect to a significant amplitude.
The particular metal in the setup, presumably iron, has a significant population of electrons with a spin that can be reoriented by an external magnetic field. Every time the current is reversed the direction of the magnetic field is reversed, and the alignable electrons realign.
However, that realignment is possible only if those electrons can exchange angular momentum with external mass. In the case of this demo the iron atoms are in a rigid structure, so the angular momentum exchanged manifests itself as angular momentum of that body of metal as a whole.
So: in this demonstration of the Einstein de Haas effect you get to see a quantum effect accumulate to a level where you see a physical consequence with the unaided eye.
In the case of a permanent magnet the spins of the aligned electrons are in a fixed orientation with respect to the crystal structure of the surrounding material. So in the case of a permanent magnet: when you reorient the magnet the spins follow that motion.
When an external magnetic field has induced spin alignment of electrons in the material those spins will remain aligned with that external magnetic field, independent of the orientation of the non-permanent magnet. (Well, there is hysteresis of the induced magnetism, but the inducing magnetic field prevails.)
About gyroscopic effect:
I don't think I have seen that question raised before:
Theoretically, with the angular momentums of the aligned electrons of a permanent magnet combined, does that imply manifestation of gyroscopic effects?
I find that an interesting question. I'm not sure. Here's a consideration: in the presence of an exernal magnetic field energy transitions of electrons are between a state of being magnetically aligned with the external field or anti-aligned (the operating principle of Magnetic Resonance Imaging). That's a non-classical two-valuedness. My best guess: I don't expect gyroscopic effects in the classical sense.