We present new experiments of particle-driven turbulent plumes issuing from a constant source of dense particle-laden fluid, with buoyancy flux, $B$, in a uniform horizontal current, $u$. Experiments show that a turbulent, well-mixed plume develops, in which the downward vertical speed $w$ decreases with depth $z$ according to $w = 0.76 (B/uz)^{1/2}$ while the horizontal speed rapidly asymptotes to the current speed $u$, provided that the Stokes settling speed of the particles $v<0.92 w$. For $v > 0.92 w$, the particles separate from the plume fluid, and their depth $z$ increases according to the simple sedimentation trajectory $\textrm {d}z/{\textrm {d}\kern0.7pt x} = v/u$. As the particles sediment, they form clusters of particles, which lead to fluctuations in the particle load with position, but do not appear to change the time-average sedimentation speed. We explore the impact of these results for deep-sea mining, in which the fate of the plume water as well as the particles is key for assessing potential environmental impacts.