The mobility mu of excess electrons has been measured in liquid mixtures of Kr and CH4 as a function of the electric field up to Eapproximate to10(4) V/cm and of the CH4 concentration x up to xapproximate to10%, at a temperature Tapproximate to125 K, fairly close to the normal boiling point of Kr (T(b)approximate to120 K) [Nucl. Instrum. Methods Phys. Res. A 430, 277 (1997)]. We present here new data extending the previous set in the low E region. At small E, mu appears to be quite independent of x. CH4 impurities prove to be less efficient in enhancing momentum transfer than liquified rare gas impurities Kr and Xe in liquid Ar. The dependence of mu on E at higher strengths is complicated. On the one hand, the addition of CH4 extends the range of E in which mu is field independent, by efficiently thermalizing the electrons. On the other hand, at the highest field, the presence of the impurities accomplishes a large increase of the electron drift velocity with respect to the pure liquid (up to a factor of 7 for the highest x). Moreover, at intermediate values of E, where electrons are epithermal, there appears to be a crossover between two different behaviors of mu as a function of E. The electric field strength at the crossover, E-star, is well correlated with x. The behavior of mu can be rationalized in terms of a gas-kinetic model proposed to explain its concentration dependence in the liquified rare gas mixtures Ar-Kr and Ar-Xe. This analysis suggests that the observed crossover is related to the excitation of the first vibrational level of CH4.