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The domain wall (DW) dynamics of amorphous and nanocrystalline Co-based glass-coated microwires are explored under the influence of stress annealing. Different annealing profiles have enabled remarkable changes in coercivity and magnetostriction values of Co-based amorphous microwires with initially negative magnitude, allowing induced magnetic bistability in stress-annealed samples and, consequently, high DW velocity has been observed. Similarly, Co-based nanocrystalline microwires with positive magnetostriction and spontaneous bistability have featured high DW velocity. Different values of tensile stresses applied during annealing have resulted in a redistribution of magnetoelastic anisotropy showing a decreasing trend in both DW velocities and coercivity of nanocrystalline samples. Observed results are discussed in terms of the stress dependence on magnetostriction and microstructural relaxation.