Extending of nanostructures into the third dimension has become a major research avenue in condensed-matter physics, because of geometry-and topology-induced phenomena. In this regard, superconducting 3D nanoarchitectures feature magnetic field inhomogeneity, non-trivial topology of superconducting screening currents and complex dynamics of topological defects. Here, we investigate theoretically topological transitions in the dynamics of vortices and phase slips in open superconductor nanotubes under a modulated transport current. Relying upon the time-dependent Ginzburg-Landau equation, we reveal two distinct voltage regimes when (i) a dominant part of the tube is in the normal or superconducting state and (ii) the complex interplay between vortices, phase slips and screening currents determines a rich FFT voltage spectrum. Our findings allow for unveiling the distributions of the superconducting order parameter in open nanotubes via recording time-dependent induced voltage and for controlling these states by using superimposed dc and ac transport currents.