You can think about a paramagnetic material as being comprised of a bunch of quantum mechanical spins aligned in random directions.
In order to better understand what is happening, I suggest that you think in parallel about the magnetic dipoles of the subatomic particles. They are randomly distributed in non-magnetic materials. Not to be misunderstood, in every atom or molecule the magnetic moments are well aligned (Paulis principle). On a larger scale of the material, these magnetic moments are neutralized in non-magnetic materials.
When an external magnetic field is applied, a fraction of the spins will align with the magnetic field. The stronger the external magnetic field, the larger the number of spins which will align with it.
Perfect. And if you think about the magnetic moments in parallel, it becomes clear why the external magnetic field makes these alignments.
We now consider an experiment in which a paramagnetic material is suspended in a vacuum, not interacting with anything. An experimenter varies an external magnetic field. As the external magnetic field is changed, the number of spins aligned with it changes as well.
Good description. One small note: The alignment is not complete. The magnetic moments of the particles that are not or not so perfectly influenced prevent other particles from reaching perfect alignment. A perfect alignment is therefore not the usual case. Furthermore, the temperature of a body (the exchange with the environment by photon emission and absorption) prevents the stable parallel alignment of the particles. The best results are therefore obtained with ultra-cooled materials.
However, quantum mechanical spins also carry angular momentum. The more the spins are aligned overall, the larger the total angular momentum. Therefore, how can the number of spins aligned with the magnetic field change if there is no way for them to transfer their angular momentum to some other object?
I prefer to describe it a little differently. The orientation of the magnetic dipole moment of the particles is a rotation, and the moment of these rotations must be compensated by another rotation. As long as the magnetic dipoles are randomly distributed, these rotations compensate each other and the sum is zero. (What you get is in any case a change in the dimensions of the body due to the different space the aligned particles need).
What would actually happen in the experiment I described? What is the process by which the spins are flipped as the magnetic field changes, and how does it not violate the conservation of angular momentum?
That my above description does not violate the conservation of angular momentum is clear when you imagine what happens if you switch off the external field.
If the material is not in a state of self-alignment (it has not been turned into a permanent magnet), the particles will return to their previous orientation, either in whole or in part. In any case, they do so in all directions (because they did so by chance during alignment) - and the angular moments compensate each other again.
If the material is converted into a permanent magnet, nothing happens.
If the particles are aligned in advance, a changing (and not parallel) external field naturally causes the body to be deflected laterally. See the answer about the Einstein-de Haas-experiment.