We investigated the properties of the interstellar medium (ISM) at giant molecular cloud (GMC) scales (~100 pc) in a sample of 27 nearby luminous infrared galaxies (LIRGs) spanning all interacting stages along the merger sequence, i.e. from isolated systems to late-stage mergers. In particular, we study the relations between star-formation (SF) and molecular gas surface density as a function of the interaction stage by (1) defining beam-sized (unresolved, line-of-sight) regions and (2) identifying actual gas clumps and physical structures within the galaxies. In total, we identify more than 4000 beam-sized CO-emitting regions defined on scales of ~100pc and more than 1000 molecular gas clumps in the sample. To map the distribution of molecular gas we used the Atacama Large Millimeter/submillimeter Array (ALMA) to observe the J=2-1 CO transition, and to map the distribution of star formation we used the Hubble Space Telescope (HST) observations of the Pa{alpha} or Pa{beta} hydrogen recombination lines. We derived spatially resolved Kennicutt-Schmidt (KS) relations for each LIRG in the sample. When using beam-sized regions, we find that 67% of galaxies follow a single relation between {SIGMA}SFR and {SIGMA}H2. However, in the remaining galaxies, the relation splits into two branches - one characterised by higher {SIGMA}SFR and {SIGMA}H2, the other by lower value - indicating the presence of a duality in this relation. In contrast, when using physical gas clumps, the duality disappears and all galaxies show a single trend. These results provide two complementary perspectives when studying the star formation process. The first maximises the number statistics (beam-sized regions), and the second focuses on actual structures associated with gas clumps in which the measured sizes have a physical meaning. We also studied other ISM and clump properties as a function of the merger stage of the LIRG systems. We find that isolated galaxies and systems in early stages of interaction exhibit smaller amounts of gas and lower star formation rates (SFRs). As the merger progresses, however, the amount of gas in the central kiloparsecs of the galaxy undergoing the merger increases, along with the SFR, and the slope of the KS relation becomes steeper, indicating an increase in the SF efficiency of the molecular gas clumps. Clumps in late-stage mergers are predominantly located at small distances from the nucleus, confirming that most of the activity is concentrated in the central regions. Interestingly, the relation between the star formation efficiency and the boundedness parameter (which measures the effects of gravity against velocity dispersion) evolves from being roughly flat in the early stages of the merger to becoming positive in the final phases, indicating that clump self-gravity only starts to regulate the star formation process between the early and mid merger stages.