We study the transport of electrons and propagation of the negative ionisation fronts in indium vapour. Electron swarm transport properties are calculated using a Monte Carlo simulation technique over a wide range of reduced electric fields E/N (where E is the electric field and N is the gas number density) and indium vapour temperatures in hydrodynamic conditions, and under non-hydrodynamic conditions in an idealised steady-state Townsend (SST) setup. As many indium atoms are in the first $(5s25p)2P3/2$ metastable state at vapour temperatures of a few thousand Kelvin, the initial Monte Carlo code was extended and generalized to consider the spatial relaxation and the transport of electrons in an idealised SST experiment, in the presence of thermal motion of the host-gas atoms and superelastic collisions. We observe a significant sensitivity of the spatial relaxation of the electrons on the indium vapour temperature and the initial conditions used to release electrons from the cathode into the space between the electrodes. The calculated electron transport coefficients are used as input for the classical fluid model, to investigate the inception and propagation of negative ionisation fronts in indium vapour at various E/N and vapour temperatures. We calculate the electron density, electric field, and velocity of ionisation fronts as a function of E/N and indium vapour temperature. The presence of indium atoms in the first $(5s25p)2P3/2$ metastable state significantly affects the characteristics of the negative ionisation fronts. The transition from an avalanche into a negative ionisation front occurs faster with increasing indium vapour temperature, due to enhanced ionisation and more efficient production of electrons at higher vapour temperatures. For lower values of E/N, the electron density behind the streamer front, where the electric field is screened, does not decay as one might expect for atomic gases, but it could be increased due to the accumulation of low-energy electrons that are capable of initiating ionisation in the streamer interior.