In glassy substances and biological media, dynamical transitions are observed in neutron scattering that manifests itself as deviations of the translational mean-squared displacement, 〈x2〉, of hydrogen atoms from harmonic dynamics. In biological media, the deviation occurs at two temperature intervals, at ∼100-150 K and at ∼170-230 K, and it is attributed to the motion of methyl groups in the former case and to the transition from harmonic to anharmonic or diffusive motions in the latter case. In this work, electron spin echo (ESE) spectroscopy - a pulsed version of electron paramagnetic resonance - is applied to study the spin relaxation of nitroxide spin probes and labels introduced in molecular glass former o-terphenyl and in protein lysozyme. The anisotropic contribution to the rate of the two-pulse ESE decay, ΔW, is induced by spin relaxation appearing because of restricted orientational stochastic molecular motion; it is proportional to 〈α2〉τc, where 〈α2〉 is the mean-squared angle of reorientation of the nitroxide molecule around the equilibrium position and τc is the correlation time of reorientation. The ESE time window allows us to study motions with τc < 10-7 s. For glassy o-terphenyl, the 〈α2〉τc temperature dependence shows a transition near 240 K, which is in agreement with the literature data on 〈x2〉. For spin probes of essentially different size, the obtained data were found to be close, which evidences that motion is cooperative, involving a nanocluster of several neighboring molecules. For the dry lysozyme, the 〈α2〉τc values below 260 K were found to linearly depend on the temperature in the same way as it was observed in neutron scattering for 〈x2〉. As spin relaxation is influenced only by stochastic motion, the harmonic motions seen in ESE must be overdamped. In the hydrated lysozyme, ESE data show transitions near 130 K for all nitroxides, near 160 K for the probe located in the hydration layer, and near 180 K for the label in the protein interior. For this system, the two latter transitions are not observed in neutron scattering. The ESE-detected transitions are suggested to be related with water dynamics in the nearest hydration shell: with water glass transition near 130 K and with the onset of overall water molecular reorientations near 180 K; the disagreement with neutron scattering is ascribed to the larger time window for ESE-detected motions.