In bond site percolation models for 'myopic ants' the jumps of a random walker are biased to attempt only open passages, whereas an unbiased walker ('blind ant') tries open and blocked ones equally. The authors generalise these concepts to impurity problems. One parameter describes the conductivity of the impurity, which may be higher or lower than that of the host sites or bonds. Another parameter describes a weak or strong bias to move away or stay close to impurities. This provides a more general description for, e.g., flow in porous media, oil in sands or gas in rocks. The models are analysed to first order in the concentration of impurities. Results are presented for the static diffusion coefficient and the velocity autocorrelation function. It is found that the bias parameter mainly enters the short- and intermediate-time regime.