We study stability of non-steady plane water evaporation fronts, arising in vertical flows with phase transition in horizontally extended domains of a porous medium. Isothermal motions in the non-wettable horizontal porous medium when a water layer is located over a vapor one is considered and homogeneous capillary forces on the phase transition front are taken into consideration. It is shown that motion of the mobile phase transition fronts depend on their initial position relative to bifurcation curves of the stationary fronts. The four domains in the parameter space are picked out where behavior of linear perturbations of the mobile fronts are different: in the first domain all perturbations decrease, in the second one only long-wave perturbations grow, in the third domain only short-wave perturbations grow and in the fourth one all perturbations grow. Dynamics of localized finite amplitude perturbations of the moving phase transition front are also studied numerically. It is established that their evolution is mainly influenced by the initial position of the perturbed front relative to the mentioned stability/instability domains. (C) 2014 Elsevier Ltd. All rights reserved.