  In this work, dynamics of chiral domain walls in long and narrow magnetic
heterostructures based on non-centrosymmetric chiral magnets with bulk
Dzyaloshinskii-Moriya interactions (DMI) and perpendicular magnetic anisotropy
is systematically investigated. The driving forces can be out-of-plane magnetic
fields and in-plane currents, correspondingly both steady and precessional
flows are considered. Their dividing points (the Walker critical field and
current density) are obtained as functions of bulk DMI strength
($D_{\mathrm{b}}$) and the ratio ($\kappa$) of total (crystalline plus shape)
anisotropy in the hard axis over that in the easy one. When far beyond Walker
breakdown, the dependence curve of wall velocity on external in-plane bias
field takes parabolic shape around the compensation point where the total
in-plane field disappears. The center shift is determined by $D_{\mathrm{b}}$,
$\kappa$, and the wall's topological charge, thus can be used to measure the
bulk DMI strength in chiral magnets.
