In this paper, we study the Cauchy problem of the classical incompressible Navier-Stokes equations and the parabolic-elliptic Keller-Segel system in the framework of the Fourier-Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier-Besov spaces for the heat equation. Such estimates are fundamental for solving certain PDE's of parabolic type. As an applications, we prove global well-posedness in variable Fourier-Besov spaces for the 3D classical incompressible Navier-Stokes equations and the 3D parabolic-elliptic Keller-Segel system.