Context. The processes responsible for galaxy evolution in different environments as a function of galaxy mass remain heavily debated. Rich galaxy clusters are ideal laboratories in which to distinguish the role of environmental versus mass quenching because they consist of a full range of galaxies and environments. Aims. Using the CLASH-VLT survey, we assembled an unprecedentedly large sample of 1234 spectroscopically confirmed members in Abell S1063. We found a dynamically complex structure at ⟨z_cl⟩ = 0.3457 with a velocity dispersion σ_v = 1380₋₃₂⁺²⁶ km s⁻¹. We investigated cluster environmental and dynamical effects by analysing the projected phase-space diagram and the orbits as a function of galaxy spectral properties. Methods. We classified cluster galaxies according to the presence and strength of the [OII] emission line, the strength of the Hδ absorption line, and colours. We investigated the relation between the spectral classes of galaxies and their position in the projected phase-space diagram. We separately analysed red and blue galaxy orbits. By correlating the observed positions and velocities with the projected phase-space constructed from simulations, we constrained the accretion redshift of galaxies with different spectral types. Results. Passive galaxies are mainly located in the virialised region, while emission-line galaxies lie beyond r₂₀₀ and are accreted into the cluster at a later time. Emission-line and post-starburst galaxies show an asymmetric distribution in projected phase-space within r₂₀₀; emission-line galaxies are prominent at Δv/σ ≲ −1.5 and post-starburst galaxies at Δv/σ ≳ 1.5, suggesting that backsplash galaxies lie at high positive velocities. We find that low-mass passive galaxies are accreted into the cluster before high-mass galaxies. This suggests that we observe as passives only the low-mass galaxies that are accreted early into the cluster as blue galaxies. They had the time to quench their star formation. We also find that red galaxies move on more radial orbits than blue galaxies. This can be explained if infalling galaxies can remain blue by moving on tangential orbits.