Fizeau drag means that the speed of light can be regulated by the flow of water, owing to the momentum interaction between photons and moving media. However, the dragging of heat is intrinsically elusive, due to the absence of momentum in thermal diffusion. Here, we design a spatiotemporal thermal metamaterial based on heat transfer in porous media to demonstrate the diffusive analog to Fizeau drag. The space-related inhomogeneity and time-related advection enable the diffusive Fizeau drag effect. Thanks to the spatiotemporal coupling, different propagating speeds of temperature fields can be observed in two opposite directions, thus facilitating nonreciprocal thermal profiles. The phenomenon of diffusive Fizeau drag stands robustly even when the direction of advection is perpendicular to the propagation of temperature fields. These results could pave an unexpected way toward realizing the nonreciprocal and directional transport of mass and energy.