  The recent discovery of magnetic topological insulators has opened new
avenues to explore exotic states of matter that can emerge from the interplay
between topological electronic states and magnetic degrees of freedom, be it
ordered or strongly fluctuating. Motivated by the effects that the dynamics of
the magnetic moments can have on the topological surface states, we investigate
the magnetic fluctuations across the
(MnBi$_{\text{2}}$Te$_{\text{4}}$)(Bi$_{\text{2}}$Te$_{\text{3}}$)$_{\text{n}}$
family. Our paramagnetic electron spin resonance experiments reveal contrasting
Mn spin dynamics in different compounds, which manifests in a strongly
anisotropic Mn spin relaxation in MnBi$_{\text{2}}$Te$_{\text{4}}$ while being
almost isotropic in MnBi$_{\text{4}}$Te$_{\text{7}}$. Our density-functional
calculations explain these striking observations in terms of the sensitivity of
the local electronic structure to the Mn spin-orientation, and indicate that
the anisotropy of the magnetic fluctuations can be controlled by the carrier
density, which may directly affect the electronic topological surface states.
