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Accretion onto a highly-magnetised neutron star runs through a magnetospheric flow, where the plasma follows the magnetic field lines in the force-free regime. The flow entering the magnetosphere is accelerated by the gravity of the star and then abruptly decelerated in a shock located above the surface of the star. For large enough mass accretion rates, most of the radiation comes from the radiation-pressure-dominated region below the shock, known as accretion column. Though the one-dimensional, stationary structure of this flow has been studied for many years, its global dynamics was hardly ever considered before. Considering the time-dependent structure of an accretion column allows us to test the stability of the existing stationary analytic solution, as well as its possible variability modes, and check the validity of its boundary conditions. Using a conservative scheme, we perform one-dimensional time-dependent simulations of an ideal radiative MHD flow inside an aligned dipolar magnetosphere. Whenever thermal pressure locally exceeds magnetic pressure, the flow is assumed to lose mass. Position of the shock agrees well with the theoretical predictions below a limit likely associated with advection effects: if more than $2/3$ of the released power is advected with the flow, the analytic solution becomes self-inconsistent, and the column starts leaking at a finite height. Depending on the geometry, this breakdown may broaden the column, mass-load the field lines, and produce radiation-driven, mildly relativistic ejecta. Evolving towards the equilibrium position, the shock front experiences damped oscillations at a frequency close to the inverse sound propagation time.