We investigate well-posedness, regularity and asymptotic behavior of parabolic Kirchhoff equations ∂tu − a (∫ |∇u|2) Δu + α(x)u = f (x) in Ω × (0, ∞), on bounded domains of RN, N ≥ 2, with non-homogeneous flux boundary conditions a (∫ |∇u|2) ∂ν∂u + β(x)u = g(x) on ∂Ω × (0, ∞) of Neumann or Robin type. The data in the problem satisfy (f, g, u(0)) ∈ L2(Ω) × L2(∂Ω) × H1(Ω). Approximated solutions are constructed using time rescaling and a complete set in H1(Ω) relating the equation and the boundary condition. Uniform global estimates are derived and used to prove existence, uniqueness, continuous dependence on data, a priori estimates and higher regularity for the parabolic problem. Existence and uniqueness of stationary solutions are shown, as well as a description about their role on the asymptotic behavior regarding to the evolutionary equation. Furthermore, a sufficient condition for the existence of isolated local energy minimizers is provided. They are shown to be asymptotically stable stationary solutions for the parabolic equation.