  We consider analytically current-driven dynamics of magnetic N\'{e}el walls
in heavy metal/ferromagnetic metal/oxide trilayers where strong spin-orbit
coupling and interfacial Dzyaloshinskii-Moriya interaction (i-DMI) coexist. We
show that field-like spin-orbit torque (FL-SOT) with effective field along
$\mathbf{n}\times\hat{\mathbf{J}}$ ($\mathbf{n}$ being the interface normal and
$\hat{\mathbf{J}}$ being the charge current direction) and i-DMI induced torque
can both lead to Walker breakdown suppression meanwhile leaving the wall
mobility (velocity versus current density) unchanged. However, i-DMI itself can
not induce the "universal absence of Walker breakdown" (UAWB) while FL-SOT
exceeding a certain threshold can. Finitely-enlarged Walker limits before UAWB
are theoretically calculated and well explain existing data. In addition,
change in wall mobility and even its sign-inversion can be understood only if
the anti-damping-like (ADL) SOT is appended. For N\'{e}el walls in
ferromagnetic-metal layer with both perpendicular and in-plane anisotropies, we
have calculated the respective modifications of wall mobility under the
coexistence of spin-transfer torque, SOTs and i-DMI. Analytics shows that in
trilayers with perpendicular anisotropy strong enough spin Hall angle and
appropriate sign of i-DMI parameter can lead to sign-inversion in wall mobility
even under small enough current density, while in those with in-plane
anisotropy this only occurs for current density in a specific range.
