Our aim is to understand the role of the environment in the quenching of star formation of galaxies located in the infall cluster region of Abell 85 (A85). This is achieved by studying the post-starburst galaxy population as tracer of recent quenching. By measuring the equivalent width (EW) of the [O II] and Hδ spectral lines, we classify the galaxies into three groups: passive (PAS), emission line (EL), and post-starburst (PSB) galaxies. The PSB galaxy population represents {˜ } 4.5 per cent of the full sample. Dwarf galaxies (Mr > -18.0) account for {˜ } 70 - 80 per cent of PSBs, which indicates that most of the galaxies undergoing recent quenching are low-mass objects. Independently of the environment, PSB galaxies are disc-like objects with g - r colour between the blue ELs and the red PAS ones. The PSB and EL galaxies in low-density environments show similar luminosities and local galaxy densities. The dynamics and local galaxy density of the PSB population in high-density environments are shared with PAS galaxies. However, PSB galaxies inside A85 are at shorter clustercentric radius than PAS and EL ones. The value of the EW(Hδ) is larger for those PSBs closer to the cluster centre. We propose two different physical mechanisms producing PSB galaxies depending on the environment. In low-density environments, gas-rich minor mergers or accretions could produce the PSB galaxies. For high-density environments like A85, PSBs would be produced by the removal of the gas reservoirs of EL galaxies by ram-pressure stripping when they pass near the cluster centre.