From the summary: A vertically moving boundary in a stratified fluid can create and maintain a horizontal density gradient, or vertical layering of density, through the mechanism of viscous entrainment. Experiments to study the evolution and stability of axisymmetric flows with vertically layered density are performed by towing a narrow fibre upwards through a stably stratified viscous fluid. The fibre forms a closed loop and thus its effective length is infinite. A layer of denser fluid is entrained and its thickness is measured by implementing tracking analysis of dyed fluid images. Thickness values of up to 70 times that of the fibre are routinely obtained. A lubrication model is developed for both a two-dimensional geometry and the axisymmetric geometry of the experiment, and shown to be in excellent agreement with dynamic experimental measurements once subtleties of the optical tracking are addressed. Linear stability analysis is performed on a family of exact shear solutions, using both asymptotic and numerical methods in both two dimensions and the axisymmetric geometry of the experiment. It is found analytically that the stability properties of the flow depend strongly on the size of the layer of heavy fluid surrounding the moving boundary, and that the flow is neutrally stable to perturbations in the large-wavelength limit. At the first correction of this limit, a critical layer size is identified that separates stable from unstable flow configurations. Surprisingly, in all of the experiments the size of the entrained layer exceeds the threshold for instability, yet no unstable behaviour is observed. This is a reflection of the small amplification rate of the instability, which leads to growth times much longer than the duration of the experiment. This observation illustrates that for finite times the hydrodynamic stability of a flow does not necessarily correspond to whether or not that flow can be realised from an initial-value problem. Similar instabilities that are neutral to leading order with respect to long waves can arise under the different physical mechanism of viscous stratification, and we draw a comparison to that scenario.