Aims: We study the dynamical status of the poor, low X-ray luminous galaxy clusters Abell 610, Abell 725, and Abell 796 (at z = 0.1, 0.09, and 0.16, respectively), containing diffuse radio sources (relic, relic, and possible halo, respectively). Methods: Our analysis is based on new spectroscopic data obtained at the William Herschel Telescope for 158 galaxies, new photometry obtained at the Isaac Newton Telescope with the addition of data recovered from the Data Release 5 of the Sloan Digital Sky Survey. We use statistical tools to select 57, 36, and 26 cluster members and to analyze the kinematics of cluster galaxies, as well as to study the 2D cluster structure. Results: The low values we compute for the global line-of-sight velocity dispersion of galaxies (σV = 420-700 km s-1) confirm that these clusters are low-mass clusters. Abell 610 shows a lot of evidence of substructure. It seems to be formed by two structures separated by ~700 km s-1 in the cluster rest-frame, having comparable σV ~ 200 km s-1 and likely causing a velocity gradient. The velocity of the brightest cluster member (BCMI; a bright radio source) is very close to the mean velocity of the higher velocity structure. A third small, low-velocity group hosts the second brightest cluster member (BCMII). The analysis of the 2D galaxy distribution shows a bimodal distribution in the core elongated in the SE-NW direction and likely associated to BCMI and BCMII groups. Abell 725 and Abell 796, which are less sampled, show marginal evidence of substructure in the velocity space. They are elongated in the 2D galaxy distribution. For both Abell 610 and Abell 725 we shortly discuss the possible connection with the hosted diffuse radio relic. Conclusions: Our results show that relic radio sources are likely connected with merger events, but are not limited to massive clusters. About the possible halo source in Abell 796, there is some evidence of a merger event in this non-massive cluster, but a pointed radio observation is necessary to confirm this halo. Tables 1, 4, 5 are only available in electronic form at http://www.aanda.org