The spreading, imbibition and solidification of a hot two-dimensional droplet on a cooler porous substrate is studied. Lubrication theory and asymptotic reduction are used to derive a coupled system of evolution equations for the droplet thickness and vertical extents of the solid/melt boundary and the saturation front within the porous medium. This medium is assumed to be a membrane composed of an array of pores having fixed width. A precursor layer model and a disjoining pressure are used to relieve the singularity at the contact line. When the solidification and imbibition time scales are similar to those associated with spreading, the dynamics follow two stages: spreading and imbibition accompanied by solidification within the pores leading to their blockage, followed by contact line arrest and basal solidification of the droplet; the possibility of crust formation at the gas–melt interface is excluded from the present study. The dependence of the dynamics on the relevant system parameters is elucidated. Furthermore, our modelling predictions compare favourably with experimental data also obtained as part of the present work.