The basic medium for implementing nonlinear switching of ultrafast optical signals is provided by dual-core optical fibers (DCF). Recently we have demonstrated that high switching contrasts at low pulse energies are offered by the new generation of all-solid highly nonlinear DCFs made of two soft glasses with sufficiently large index difference. The experiments have confirmed general predictions of previous theoretical works, and simulations of an appropriate model of the DCF have produced results which accurately agree with the experimental findings . Following the successful analysis of the system with symmetric cores, here we extend the work, taking into account the propagation-constant mismatch between the cores . The experimental and numerical results reveal that the asymmetry improves the switching performance of the nonlinear couplers based on DCFs, in the cases of both self-switching of near-IR femtosecond pulses and cross-switching controlled by the pulses at a shorter wavelength.