This work analyses the viscous flow and elastic deformation created by the forced axial motion of a rigid cylinder within an elastic liquid-filled tube. The examined configuration is relevant to various minimally invasive medical procedures in which slender devices are inserted into fluid-filled biological vessels, such as vascular interventions, interventional radiology, endoscopies and laparoscopies. By applying the lubrication approximation, thin shell elastic model, as well as scaling analysis and regular and singular asymptotic schemes, the problem is examined for small and large deformation limits (relative to the gap between the cylinder and the tube). At the limit of large deformations, forced insertion of the cylinder is shown to involve three distinct regimes and time scales: (i) initial shear dominant regime, (ii) intermediate regime of dominant fluidic pressure and a propagating viscous-peeling front, (iii) late-time quasi-steady flow regime of the fully peeled tube. A uniform solution for all regimes is presented for a suddenly applied constant force, showing initial deceleration and then acceleration of the inserted cylinder. For the case of forced extraction of the cylinder from the tube, the negative gauge pressure reduces the gap between the cylinder and the tube, increasing viscous resistance or creating friction due to contact of the tube and cylinder. Matched asymptotic schemes are used to calculate the dynamics of the near-contact and contact limits. We find that the cylinder exits the tube in a finite time for sufficiently small or large forces. However, for an intermediate range of forces, the radial contact creates a steady locking of the cylinder inside the tube.