Kinetic processes, which are joined with mass transport, are studied in the vicinity of the sharp LCST-type demixing transition of semi-dilute aqueous poly(N-isopropyl acrylamide) (PNIPAM) solutions. These processes are slow as compared to the highly cooperative collapse of individual polymer chains. Purely elastic properties, that are particularly sensitive to this phase transition, are addressed depending on the temperature, space and time by Brillouin spectroscopy. Above the demixing temperature T-c, we discriminate between kinetics related to the phase separation into PNIPAM-rich and PNIPAM-poor domains and kinetics connected to the impact of gravitation on the on-going phase separation. Using shallow temperature jumps of 0.3 degrees C, the growth of compact PNIPAM-rich agglomerates with identical gel-like mechanical consistency is provoked independently of temperature and position within the sample above T-c. Astonishingly, the transition temperature does not vary while heating or cooling the solutions across the phase transition, although the elastic properties depend strongly on space and time during the equilibration of PNIPAM concentration gradients following the re-entrance into the low-temperature phase.