Broadband dielectric spectroscopy, positron annihilation lifetime spectroscopy and differential scanning calorimetry are used to investigate the molecular dynamics, free volume and thermal behaviour of a poly(ethyleneimine), PEI, canopy in liquid-like nanoparticle-organic hybrid materials (NOHMs) consisting of alumina nanoparticles and nanorods as inorganic nanocores. It is confirmed that the highly branched PEI canopy in liquid-like NOHMs possesses an ordered structure having less entanglements of the side chains as compared to neat PEI. The size of the free volumes associated with the PEI canopy for nanoparticle- and nanorod-based NOHMs is larger and smaller, respectively, as compared to neat PEI. The time scales characterizing the segmental dynamics and side chain relaxations of the canopy in the nanorod-based NOHM are slowed down drastically as compared to the nanoparticle-based NOHM and neat PEI. These results confirm that the shape of the inorganic cores plays a deterministic role toward the structural arrangement (free volume) and segmental dynamics of the canopy polymer in liquid-like NOHMs.