Based on the almost fibril state of magnetic flux in astrophysical bodies we present a study of the stability of magnetic flux tubes which are described by the thin flux tube approximation. A stationary flow is allowed in the tube which in equilibrium lies in a vertical plane, while the surroundings are assumed to be static. This complements earlier studies with external but no internal flows. The equations governing small adiabatic displacements are given in a canonical form which allows the proof that operators are Hermitian. This form is derived for the first time. The character of operators allows powerful conclusions about the eigensolutions, sufficient conditions for stability as well as the application of variational techniques. The velocity may potentially lead to the phenomenon of overstability. Special cases are discussed analytically. These cover the local approximation, binormal perturbations, static equilibria, and horizontal and vertical flux tubes. Stability of siphon flows along arch-shaped flux tubes describing the Evershed effect in penumbrae of sunspots is analyzed numerically. A standard model is found to be stable. Increasing the flow speed or decreasing the magnetic field strength however leads to monotonic instability. Further astrophysical applications are mentioned.