The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The nonlinear stability of the basic flow is analysed theoretically at asymptotically large Reynolds number and it is found that non-axisymmetric finite-amplitude neutral modes can be supported over a wide range of thread radii and injection velocities. The axial force on the thread is calculated and it is found to be significantly less than that predicted by undisturbed-flow theory, in agreement with thread–annular experiments.