Vortex beams have garnered significant attention in cutting-edge fields such as modern optical communications, quantum information, and micro-nano manipulation due to their unique orbital angular momentum (OAM). However, the propagation characteristics of phase singularities in various complex optical fields remain a core research focus and technical challenge. Based on the paraxial wave equation, this paper establishes a general analytical propagation model for the phase singularities of noncanonical vortices embedded in arbitrary complex host beams. This model addresses the evolution characteristics of the spatial trajectory curves and shape factors of propagating non-canonical optical vortices in the free space. It is pointed out that due to the intensity and phase inhomogeneity of the complex host beam, the propagation of phase singularities is generally accompanied by the dynamic evolution of ellipticity and rich topological events, such as self-induced precession, oscillation, and annihilation between vortex pairs. Regarding the influence of the spatial distribution of the host beam on the trajectory of off-axis vortex points, studies show that the intensity gradient drives the phase singularity to move in the perpendicular direction. On the other hand, the phase gradient guides the vortex point to move in a direction parallel to the gradient. Particularly, the proposed analytical model has been applied to explain the inversion of topological charge successfully, which seems to violate the principle of topological charge conservation during free space propagation without external perturbations. The proposed model systematically interprets this striking phenomenon during the propagation of off-axis noncanonical vortex in asymmetric light beams, reveals the conditions for the inversion, predicts the propagation trajectories of phase singularities, and explains the physical mechanism underlying this effect from the perspective of transversal energy flow. Research indicates that the exchange between the OAM of non-canonical vortex beams and the local OAM of the asymmetric background beams constitutes the underlying physical mechanism for topological charge inversion. The non-canonical vortex core gains or loses OAM from its host beam, resulting in phase variations of its shape factor A and even polarity reversals. Nevertheless, the total OAM of the entire beam remains conserved, and this process does not alter the absolute value of the topological charge l of the vortex beam. The free space propagation characteristics of vortex points stand as one of the most central and fundamental research topics in the field of structured light fields, these research findings hold important theoretical significance and practical potential for the precise manipulation of structured optical fields and the development of novel optoelectronic devices based on orbital angular momentum.