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The temperature-dependent dynamics of the hydrogen/deuterium atoms in geometrically frustrated magnets Co₂(OH)₃Br and its deuterated form Co₂(OD)₃Br were investigated by muon spin relaxation (<i>&#956;</i>SR). The deuterium atoms in Co₂(OD)₃Br were found to be rapidly fluctuating at high temperatures, which should be arising as a quantum atomic effect due to the small mass of deuterium, then they drastically slowed down toward <i>T_c</i> = 250 K where a broad anomaly appeared in the dielectric response, and finally became quasi-static at around 180 K. Meanwhile, the hydrogen atoms in Co₂(OH)₃Br also exhibited a two-step slowing at ~240 K and ~180 K, respectively. The revealed properties in Co₂(OH)₃Br/Co₂(OD)₃Br are reminiscent of relaxor-type ferroelectrics. The present study suggested the effectiveness of the <i>&#956;</i>SR technique on revealing the hydrogen/deuterium (H/D) dynamics in Co₂(OH)₃Br/Co₂(OD)₃Br. Furthermore, magnetic coupling was found to be existing at high temperatures in this system. This work provides clear evidence to the mechanism of ferroelectric responses in the hydroxyl salts, i.e., the slowing of protons (deuterium ions) is directly related to the newly revealed ferroelectricity.