The equilibrium of non-force-free twisted horizontal magnetic flux tubes is studied including gravity and an arbitrary pressure perturbation on the tube boundary. To solve this free-boundary problem, we use general nonorthogonal flux coordinates and consider the two-dimensional case in which there is no variation of the physical quantities along the tube axis. For the applications in the convection zone and corona, we consider the case of weak external stratification by assuming that the radius of the tube is smaller than the external pressure scale height. This allows us to introduce a perturbation scheme which is much less restrictive than the customary slender flux-tube approximation. In particular, it has the advantage of not imposing any limitation on the strength of the azimuthal field as compared to the longitudinal field. Within this scheme, one retains to zero order all the functional degrees of freedom of a general axisymmetric magnetostatic equilibrium; the geometry of the perturbed azimuthal field lines is then obtained from the equilibrium equations as a consequence of the zero-order density (or rather buoyancy) distribution in the tube and of the circular wavenumber of the external pressure perturbation. We show that, as a result of the presence of gravity, the field lines are no longer concentric, although they continue being circular. The resulting changes in magnetic pressure and tension of the azimuthal field exactly counteract the differences in buoyancy in the tube cross section. On the other hand, external pressure fluctuations of circular wavenumber higher than one can only be countered by bending the azimuthal field lines. In general terms, the present scheme allows one to study in detail the mutual dependence of the (differential) buoyancy in the tube, the intensity and field line geometry of the azimuthal magnetic field, and the gas pressure and longitudinal magnetic field distributions. The main application of the equations and results of this paper is to study the transverse structure of magnetic flux rings embedded in a stratified medium with a flow around the tube that causes pressure fluctuations on its surface. This includes tubes in the deep convection zone, e.g., in its subadiabatic lower part, or those kept in place by a meridional flow. It also applies to flux rings moving in a quasi-static regime in which the drag force of the relative motion with respect to the external medium exactly compensates the total buoyancy of the tube. In this way, this study can complement the numerical simulations of the rise of magnetized tubes and bubbles toward the surface.