Force on a moving charge: $\vec{F} = q\vec{v} \times \vec{B}$. Force on a current element: $d\vec{F} = Id\vec{l} \times \vec{B}$. The connection: current $I = nAqv_d$, so $Id\vec{l} = nAqv_d dl = nA dl \cdot q\vec{v}_d$. Both are manifestations of the Lorentz force. The magnetic force never does work on a charged particle (always perpendicular to velocity), but it CAN do work on a current-carrying wire (because internal energy of the battery is converted).