The combined effect of magnetic field and current on domain wall motion is investigated in epitaxial [Co/Ni] microwires. Both thermally activated and flow regimes are found to be strongly affected by current. All experimental data can be understood by taking into account both adiabatic and nonadiabatic components of the spin transfer torque, the parameters of which are extracted. In the precessional flow regime, it is shown that the domain wall can move in the electron flow direction against a strong applied field, as previously observed. In addition, for a large range of applied magnetic field and injected current, a stochastic domain wall displacement after each pulse is observed. Two-dimensional micromagnetic simulations, including some disorder, show a random fluctuation of the domain wall position that qualitatively matches the experimental results.