Direct numerical simulations of a statistically stationary spatially decaying turbulence were performed to study the dispersion of evaporating droplets in a non-homogeneous flow. High-order finite-difference methods have been used to solve the compressible gas equations along with a Lagrangian solver for the polydispersed spray to produce a reliable reference solution. The key and novel point in this paper is to provide a comprehensive analysis from an Eulerian point of view of the turbulent dispersion of evaporating polydispersed sprays with various mean Stokes numbers. For this purpose, our work focuses on simplified evaporation laws and one-way coupling to isolate some of the key physical phenomena to be captured by an Eulerian description, which would be hidden by the intricacy of the numerous couplings occurring with more complex models. A special emphasis is laid on the polydispersed character of the spray in the particular chosen configuration. It is shown that the dynamics of the droplets at the given streamwise directions are conditioned by their size. It appears that the dispersion follows a Gaussian behaviour but around the conditioned droplets' mean velocity and not around the mean velocity of the carrier phase, both of them being distinct. A last set of analyses is dedicated to the characteristics of the carrier phase sampled by the dispersed phase. The dependence of droplet behaviour on the Stokes number is detailed