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The outward propagation of asymmetries introduced to originally axisymmetric turbulent flows is investigated, where 3D Batchelor vortices at high Reynolds numbers and with arbitrary swirl numbers are chosen as test cases. It is well known that disturbances (asymmetries) added to a 2D axisymmetric flow propagate radially outward, in order to re-axisymmetrize the vortex, but cease to travel at a critical distance, known as the stagnation radius. We utilize direct numerical simulations (DNS) and an inviscid model developed by linearizing the momentum and vorticity transport equations around the base (unperturbed) flow in helical coordinates to demonstrate that, in contrast with 2D cases, 3D vortices enable the unbounded radial propagation of asymmetries. We further apply the Wenzel-Kramers-Brillouin (WKB) analysis to the linear model, treating perturbations as compact wavepackets, to transform the PDEs of the linear model to a few ODEs. However it has been shown in the climate science community that the WKB approach ubiquitously predicts stagnation radii and heights for disturbances introduced to 3D cyclone-like vortices, here, we identify a narrow range of parameters, for which the WKB analysis also supports an unrestrained propagation for disturbances. Finally, the mechanisms governing the momentum transport at different stages and locations, thereby promoting the outward advection of perturbations, are elucidated using the the linear rapid distortion theory and numerical simulations. Since the DNS of the full Navier-Stokes equations rapidly stabilizes to a laminar configuration, the DNS of the linearized transport equations and the nonlinear governing equations without base-flow interactions are also conducted, in order to uncover the primary mechanisms for the growth and radial propagation of perturbations as well as the nonlinear processes causing growth arrest at fixed swirl numbers.