The magnetic nature of the CaFe₂O₄-type NaMn₂O₄ and Li_0.92Mn₂O₄, in which Mn ions form a zigzag chain along the b-axis, have been investigated by muon-spin rotation and relaxation μ⁺SR) and susceptibility (χ) measurements in the temperature range between 1.9 and 250 K using polycrystalline samples. Weak transverse-field μ⁺SR measurements revealed the existence of a bulk antiferromagnetic (AF) transition for NMO at T_N = 39 K and for Li_0.92Mn₂O₄ at T_N = 44 K, although the χ(T) curve did not exhibit a clear anomaly around T_N for either compound. Below T_N, however, the zero-field μ⁺SR spectra in both samples did not show the oscillatory signal characteristic of long-range magnetic order. Instead, a slowly relaxing tail is observed. Comparisons with the calculated internal magnetic field for the AF ordered phase suggests that the AF spin structure is disordered and/or rapidly fluctuating. In the paramagnetic region above 100 K, a nuclear magnetic field in Li_0.92Mn₂O₄ was found to be dynamic, whereas that in NaMn₂O₄ was almost static. This implies the diffusive motion of Li⁺ ions in Li_0.92Mn₂O₄ even below 250 K.