What kind of air flow takes place in the middle of a tornado, it is difficult to say, since reliable direct experimental observations there. It is likely that there are vertical flow, but hardly very strong.
In general, the famous lift a tornado is not due to the fact that the vortex sucks a subjects, and because of the fact that the rotating column of air has a vertical turbulence. That is, the air encircles the funnel is not strictly a circle (if there was no movement to vertical) and a fixed spiral (there would have been a constant vertical motion) and has a rapidly-changing vertical velocity component.
As for the problem of US tornadoes - one of the most important meteorological problems, their study regularly allocated considerable money. It is planned to create even a heavy machine-type tank, which would be both very difficult and quite nimble. Such a machine would be able to catch up with the tornado, enter his center and spend the experiments. So it is necessary to have patience: perhaps a year or two we'll know about the internals of a tornado is much larger than it is now.