What is the Lorentz's force equation for a moving segment of current-carrying wire?

According to Griffths’s Electrodynamics textbook, the magnetic force on a segment of current-carrying wire is

$$mathbf{F} = int I (dmathbf{l} times mathbf{B}) $$, when the path of the flow is dictated by the shape of the wire, that is $dmathbf{l}$ have the same direction as $mathbf{v}$. My question is what happen when the segment of current-carrying wire is moving, which means that the velocity vector $mathbf{v}$ of the charge has a component along the direction of $dmathbf{l}$ and also a component in the direction of the velocity vector $mathbf{V}$ of the wire. In that case, the magnetic force is (returning to the basics) the following?

$$mathbf{F} = int [(mathbf{v}+ mathbf{V})times mathbf{B}] lambda dl$$

$$mathbf{F} = int I (dmathbf{l} times mathbf{B}) + int (mathbf{V}times mathbf{B}) lambda dl$$
, where $lambda$ is the charge density.

If it is true, then when the dipole moment $mathbf{m}$ is defined using $mathbf{F} = int I (dmathbf{l} times mathbf{B}) $, then torque $mathbf{N} = mathbf{m} times mathbf{B}$ is calculated for the instant when the current loop is at rest?