We develop a homogenization theory for a spatiotemporally modulated wire medium. We first solve for the modal waves that are supported by this composite medium, and we show peculiar properties such as extraordinary waves that propagate at frequencies below the cutoff frequency of the corresponding stationary medium. We explain how these unique solutions give rise to an extreme Fresnel drag that exists already with weak and slow spatiotemporal modulation. Next we derive the effective material permittivity that corresponds to each of the first few supported modes, and write the average fields and Poynting vector. Nonlocality, nonreciprocity, and anisotropy due to the spatiotemporal modulation are three inherent properties of this medium, and are clearly seen in the effective material parameters. Lastly, we validate that homogenization and spatiotemporal variation are not necessarily interchangeable operations. Indeed, in certain parameter regimes the homogenization should be performed directly on the spatiotemporally modulated composite medium, rather than the stationary medium being homogenized first and then the effect of the space-time modulation being introduced phenomenologically.