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The rapid scale broadening and divergence increasing of vortex beams (VBs) with orbital angular momentum (OAM), e.g., Laguerre-Gaussian beams, severely impede the wide applications of VBs ranging from optical manipulation to high-dimensional quantum information communications, which call for VBs to have the same transverse scale and divergence for distinct OAM or even the small vortex ring for large OAM. Non-diffracting beams, on the other hand, that are capable of overcoming diffraction without divergence, are very evocative and indeed appealing in numerous applications including atom optics and medical imaging. Here, we propose theoretically and demonstrate experimentally a brand new type of VB having OAM-independent radii meanwhile holding propagation-invariant without divergence as well as self-healing properties, named non-diffracting perfect vortex beam (NDPVB). We work out a versatile toolkit based on Fourier-space analysis to multidimensionally customize NDPVBs at will so that it is of propagating intensity and phase controllability with intriguing customizable behaviors of self-accelerating, self-similar, and self-rotating. This goes beyond tailoring the transverse plane to the higher-dimensional propagating characteristics in structured light beams. A deeper insight into the internal flow revealed and confirmed that the multidimensional customization of NDPVBs is dominated by inducing corresponding multidimensional internal flow, facilitating our understanding of how our design scheme of propagating properties manipulates the internal flows, unveiling the nature of structure formation and behavior transformation of structured light beams.