Effects of interfacial Dzyaloshinskii-Moriya interaction (IDMI) on current-driven dynamics of ${180}^{\ensuremath{\circ}}$ domain walls (180DWs) in long and narrow spin valves (LNSVs) with heavy-metal cap layers are investigated. We focus on LNSVs with in-plane magnetic anisotropy in their free layers. For planar-transverse polarizers (pinned layers of LNSVs), the Walker breakdown is postponed considerably (practical infinity) by IDMI. More interestingly, the originally unstable traveling mode is stabilized by IDMI regardless of the polarization magnitude while bearing a high saturated velocity and thus can serve as a fast carrier of information. For parallel polarizers, the Walker limit is increased, and the corresponding modifications of IDMI to wall velocity in both steady and precessional flows of 180DWs are provided. As for perpendicular polarizers, precessional flow of 180DWs is absent due to the stability of the stationary mode beyond the modified Walker limit. Similar results are obtained for LNSVs with perpendicular magnetic anisotropy in their free layers. Our findings open possibilities for developing magnetic nanodevices based on 180DW propagation with low energy consumption and high robustness.