The flux tube being conductive (because filled by plasma) a perpendicular current will arise due to the v x B electric field. the current is of the order of 4 (v X B)/B R Sqrt(rho_m/mu_0) (For non collisional plasma, you have to take Alfvén wave conductance and possibly add a geometrical a coefficient). Then, a J x B force develops that conveniently forces the flux tube to move with the background flow.  Obviously, the problem is symmetrical, and the background flow close to the tube sees a similar force which decelerates it, like a neutral fluid encountering a cylinder.

I presume that by "background flow of ionized plasma" you mean the flow of the plasma surrounding your flux tube, which is also threaded by a frozen-in field but with a different topology, so that your flux tube is like a plasmoid.

One important consideration is whether your background plasma has a pressure gradient. If it does then the ions and electrons have opposite drift velocities and in steady state the plasmoid will generally flow at a velocity intermediate between that of the ions and electrons, depending on its size compared to the ion Larmor radius. Another consideration is the propagation velocity of the tube, since it can excite wakes that exert forces on the tube, just like the wakes of ships.