  We discuss the emergence of an effective low-energy theory for the real-time
dynamics of two classical impurity spins within the framework of a prototypical
and purely classical model of indirect magnetic exchange: Two classical
impurity spins are embedded in a host system which consists of a finite number
of classical spins localized on the sites of a lattice and interacting via a
nearest-neighbor Heisenberg exchange. An effective low-energy theory for the
slow impurity-spin dynamics is derived for the regime, where the local exchange
coupling between impurity and host spins is weak. To this end we apply the
recently developed adiabatic spin dynamics (ASD) theory. Besides the
Hamiltonian-like classical spin torques, the ASD additionally accounts for a
novel topological spin torque that originates as a holonomy effect in the
close-to-adiabatic-dynamics regime. It is shown that the effective low-energy
precession dynamics cannot be derived from an effective Hamilton function and
is characterized by a non-vanishing precession frequency even if the initial
state deviates only slightly from a ground state. The effective theory is
compared to the fully numerical solution of the equations of motion for the
whole system of impurity and host spins to identify the parameter regime where
the adiabatic effective theory applies. Effective theories beyond the adiabatic
approximation must necessarily include dynamic host degrees of freedom and go
beyond the idea of a simple indirect magnetic exchange. We discuss an example
of a generalized constrained spin dynamics which does improve the description
but also fails for certain geometrical setups.
