Do you know an example of a process that moves angular momentum from one kind of spin to the other?
Spin being an intrinsically quantum mechanical concept, I'm afraid the microscopic mechanism by which that transfer occurs will only be explainable in a quantum mechanical context. Here it will appear as a term in the Hamiltonian coupling the spin of an electron to its motion in a potential.
https://en.m.wikipedia.org/wiki/Einstein%E2%80%93de_Haas_eff...