We propose a homotopy continuation method called FLUX for approximating complicated probability density functions. It is based on progressive processing for smoothly morphing a given density into the desired one. Distributed ordinary differential equations (DODEs) with an artificial time $\gamma \in [0,1]$ are derived for describing the evolution from the initial density to the desired final density. For a finite-dimensional parametrization, the DODEs are converted to a system of ordinary differential equations (SODEs), which are solved for $\gamma \in [0,1]$ and return the desired result for $\gamma=1$. This includes parametric representations such as Gaussians or Gaussian mixtures and nonparametric setups such as sample sets. In the latter case, we obtain a particle flow between the two densities along the artificial time. FLUX is applied to state estimation in stochastic nonlinear dynamic systems by gradual inclusion of measurement information. The proposed approximation method (1) is fast, (2) can be applied to arbitrary nonlinear systems and is not limited to additive noise, (3) allows for target densities that are only known at certain points, (4) does not require optimization, (5) does not require the solution of partial differential equations, and (6) works with standard procedures for solving SODEs. This manuscript is limited to the one-dimensional case and a fixed number of parameters during the progression. Future extensions will include consideration of higher dimensions and on the fly adaption of the number of parameters.