In this work, we studied a pressure-driven flow of a magnetorheological suspension through a cylindrical tube in the presence of a nonuniform magnetic field perpendicular to the tube and varying along its axis. The flow was realized with the help of a commercial capillary rheometer in a controlled-velocity mode. Experimental pressure-flow rate curves exhibited a local minimum, and flow instabilities were observed in the range of flow rates corresponding to the decreasing branch of these curves. The nonmonotonic behavior of the flow curves is attributed to the interplay between the hydrodynamic dissipation and the interaction between particle aggregates and walls. Our theoretical model, based on the particle flux conservation, correctly predicts the shape of the pressure-flow rate curves and indicates the speed range within which flow instabilities are expected. These instabilities are manifested by somewhat regular oscillations of the pressure difference and of the outlet flow rate at a constant imposed piston speed. Visualization of particle structures in a transparent tube revealed that the flow oscillations were governed by both the suspension compressibility and the stick-slip of the aggregates on the tube walls. This study is motivated by the problem of particle clogging in magnetorheological smart devices employing nonuniform magnetic fields. (C) 2013 The Society of Rheology.