In magnetic rings of transition-metal ions, electron-spin correlations are found to exhibit rather unusual dynamical features, as a result of a subtle interplay between finite-size effects and magnetic interactions distinctly smaller than the dominant Heisenberg exchange Hamiltonian. These findings, mostly obtained by a mode-coupling approach, are shown to have important consequences on the nuclear-spin (proton) relaxation rates T-1(-1) as measured in two specific rings, comprising six Cu or six Fe ions (respectively, ferromagnetic and antiferromagnetic). In both systems the magnetic-field and temperature dependence of the theoretical results is found to compare rather well with the experimental findings.