Accelerated bodies immersed in a fluid experience enhanced inertia due to the associated co-acceleration of a certain volume of fluid in their environment. We discuss the concept of enhanced inertia in the framework of the approximation of thin flux tubes, which is widely used to describe the dynamics of concentrated magnetic structures in astrophysical objects. Previous attempts to incorporate this effect have used a local approach, in which the reaction force of the external medium on a given tube mass element solely depends on the relative acceleration of tube and environment at that element. We show that those previous formulations are inconsistent (either on physical or geometrical grounds). We present here an alternative derivation of the enhanced inertia term by geometrical means, still within a local treatment of the problem but avoiding the pitfalls of previous works. Our formulation, on the other hand, reveals a basic problem: all local approaches are bound to give incorrect answers for the reaction force in as far as they disregard the variation of the external flow in the direction parallel to the flux tube: in doing so, they generally fail to provide for global momentum conservation. An exact solution and detailed analysis for an instance of this failure is given. The discussion of this paper may be of use also in the hydrodynamical framework of vortex tube dynamics.