Monolayers of ligand-grafted nanoparticles at fluid interfaces exhibit a complex response to deformation due to an interplay of particle rearrangements within the monolayer, and molecular rearrangements of the ligand brush on the surface of the particles. We use grazing-incidence small-angle X-ray scattering (GISAXS) combined with pendant drop tensiometry to probe in situ the dynamic organization of ligand-grafted nanoparticles upon adsorption at a fluid-fluid interface, and during monolayer compression. Through the simultaneous measurements of interparticle distance, obtained from GISAXS, and of surface pressure, obtained from pendant drop tensiometry, we link the interfacial stress to the monolayer microstructure. The results indicate that, during adsorption, the nanoparticles form rafts that grow while the interparticle distance remains constant. For small-amplitude, slow compression of the monolayer, the evolution of the interparticle distance bears a signature of ligand rearrangements leading to a local decrease in thickness of the ligand brush. For large-amplitude compression, the surface pressure is found to be strongly dependent on the rate of compression. Two-dimensional Brownian dynamics simulations show that the rate-dependent features are not due to jamming of the monolayer, and suggest that they may be due to out-of-plane reorganization of the particles (for instance expulsion or buckling). The corresponding GISAXS patterns are also consistent with out-of-plane reorganization of the nanoparticles.