We report on the influence of graded magnetic anisotropy designed by stress-annealing of magnetic microwire at variable annealing temperature on domain wall propagation. We found that the domain wall propagation in a medium with graded magnetic anisotropy is substantially nonuniform. Domain wall can be trapped in the microwire region with strong enough stress-annealing induced magnetic anisotropy. On the other hand, faster domain wall propagation and a decrease in the domain wall length are observed in the region with moderate stress-annealing induced magnetic anisotropy. Beneficial effect of stress-annealing on the domain wall dynamics is associated with the induced transverse magnetic anisotropy in the outer domain which affects the travelling domain wall in a similar way as application of transversal bias magnetic field. Observed decreasing of the half-width of the electromagnetic force (EMF) peak in stress-annealed microwires can be associated to the decreasing of the characteristic domain wall length.