Results on the investigation of the propagating 180deg domain walls in bistable amorphous glass-coated microwires with nearly zero magnetostriction are reported for the first time. As-cast glass-coated microwires are bistable only if their metallic nucleus diameter is larger than 20 mum. Glass removal induces bistability in microwires with metallic nucleus diameters below 20 mum. Nearly zero magnetostrictive glass-coated microwires display larger domain wall velocities and mobilities as compared to positive magnetostrictive microwires. Samples that become bistable after glass removal display smaller values of the wall mobility as compared to as-cast bistable microwires. Mobility can be increased by annealing. The experimental results have been explained based on the damping mechanisms of the domain wall motion, specifically on the spin relaxation damping, whose coefficient is proportional to the anisotropy constant from the microwire's inner core. Stress relief determined by glass removal and annealing have been considered. The results are important for future applications of nearly zero magnetostrictive microwires in spintronic devices.