Dual active galactic nuclei (AGNs) at kpc scales that originated from galaxy mergers have been found, but the dynamics of the mergers are insufficiently understood, especially the relevant observational consequences. We present evolutionary models of dual AGNs in two cases driven by different friction laws with their surroundings, which are characterized by forces with F∝V* and . The evolutionary behaviours show that there is a minimum time-scale of ∼108 yr to accomplish a merger. Since the total friction and moment acting on dual AGNs with unequal masses do not vanish, the centre of mass moves with a velocity of Vc∼ a few ten km s−1 and makes an offset of ΔS∼ sub-kpc of the merged AGNs with respect to the centre of their host galaxies. The relative motion of the two cores along the line joining the centre of mass is of VR∼ a few ten km s−1, which determines the merger time-scales. Dissipation of the orbital kinetic energy of dual AGNs either enhances stellar velocity dispersion or heats up the interstellar medium in its revirialization to form a soft X-ray cocoon enshrouding them with a typical luminosity of ∼1042 erg s−1. We predict that galaxies with offset AGNs are likely to host binary supermassive black holes at sub-kpc scales, shedding light on searching candidates from AGNs.