Context. Moving structures have been detected in coronal bright points and in a solar flare in active regions, which were bi-directional, symmetrical, simultaneous, and quasi-periodic (Ning & Guo 2014;Ning 2016;Li et al. 2016a). They could be regarded as observational evidence of plasma outflows via magnetic reconnection. Aims. We explored pairs of moving structures in fifteen ultraviolet bright points (UBPs), which were observed in the quiet Sun or inside a small active region on 19 November 2020. Methods. The UBPs were measured by the High Resolution (HRI) Telescopes of the Extreme Ultraviolet Imager (EUI) on board the Solar Orbiter (SolO) in two passbands, HRI EUV 174 Å and HRI Lyα 1216 Å. The pairs of moving structures are identified in timedistance slices along curved slits of UBPs, and their quasi-periods are determined from the fast Fourier transform and wavelet analysis method.Results. Moving structures observed in ten UBPs as starting from their bright cores and propagating toward two ends, are interpreted as diverging motions of bi-directional moving structures. These moving structures are also characterized by simultaneity and symmetry and in the case of seven UBPs they exhibit quasi-periodicity. Moving structures seen in another five UBPs as originating from double ends and moving closer, and merging together, are manifested as converging motions. A sympathetic UBP induced by the primary UBP is observed at the edge of a small active region, and their moving structures also show the converging motion. Conclusions. The diverging motions of bi-directional moving structures could be generated by outflows after magnetic reconnections. The converging motions of two moving structures might be caused by inflows through the magnetic reconnection, or might be interpreted as upflows driven by the chromospheric evaporation.