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We study the propagation of non-isotropically correlated vector vortex beams (NCVVBs) through an ABCD optical system. The statistical properties, namely, intensity, state of polarization (SoP), degree of polarization (DoP) and degree of coherence (DoC) of these beams are investigated. The terminology non-isotropically correlated (NC) is used for electromagnetic Gaussian Schell-model (EGSM) beams, that exhibit the distinguishability of spatial correlations between parallel and orthogonal electric field components. The non-isotropic feature of the correlation widths introduces an azimuthal asymmetry in the intensity, DoP, and DoC distributions. The azimuthal asymmetry in the intensity and DoC distributions become prominent around the focal plane. However, this asymmetry can be clearly observed in DoP distributions even at a very short propagation distance. It is found that the statistical properties of NCVVBs are dependent on both Poincar´e-Hopf index (PHI) and the source correlation parameters. The number of beamlets in the intensity distribution is twice the magnitude of the PHI of the input beam. Unlike isotropically correlated vector vortex beams (ICVVBs), correlation-induced polarization around the central core of the NCVVB is observed. The DoC distribution exhibits the evolution of correlation singularities in the form of dislocations at far field plane. These dislocations in the DoC under lower correlations are dependent on the PHI, which provide a feasible approach to measure index of NCVVBs. This study provides a technique to synthesize beams with structured correlation and polarization features.