The recently developed Particle-based Variational Inference (ParVI) methods drive the empirical distribution of a set of fixed-weight particles towards a given target distribution by iteratively updating particles' positions. However, the fixed weight restriction greatly confines the empirical distribution's approximation ability, especially when the particle number is limited. In this paper, we propose to dynamically adjust particles' weights according to a Fisher-Rao reaction flow. We develop a general Dynamic-weight Particle-based Variational Inference (DPVI) framework according to a novel continuous composite flow, which evolves the positions and weights of particles simultaneously. We show that the mean-field limit of our composite flow is actually a Wasserstein-Fisher-Rao gradient flow of the associated dissimilarity functional. By using different finite-particle approximations in our general framework, we derive several efficient DPVI algorithms. The empirical results demonstrate the superiority of our derived DPVI algorithms over their fixed-weight counterparts.