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Here is what I think is happening:
For the case of H2 it is constanly being made in certain amounts by the Earths Oceans,Polar caps, and fresh water lakes. Now as the H2 rises up through all the turbulent layers of the atmospere it is readly combining with O2 and other elements because it is a hungry molecule. What doesn't get a chance to combine with other elements eventually makes it to space. It is a small amount. It reaches a balance so to say.
If a very very large pocket of natural gas were to bubble up from the ocean floor, I would expect to see an increase of the H2 leaving the atmosphere and going into space. There is nothing for it to combine with anymore. The O2 in the atmosphere has become saturated with the H2, and what is left escapes to space. It is not bound to Earth by gravity.It is bound to Earth by chemical reaction making heavier elements.
For the case of Helium there isn't that much of it. It doesn't combine so eventually it makes it out also to space. But, it takes twice as long because it is twice as heavy as H2.
So, by adding close to 100% H2 in the tube I would also expect to see it rise up through the tube and into space.

So if the effect of gravity is so small over low altitudes, why would the effect be so great over high altitudes?

I agree if it was an enclosed system. However, it has a place to escape out the other end.The more molecules you add to the system the more the system pressure will increase. Thus, since it is open at the end, and that end is near a perfect vacuum in space, it will flow.

Any increase at the bottom should send H2 out the top. Gravity does not confine H2 to our atmosphere.So why would it inside the tube?

Field Marshal Mathers,
So where would be the best place to install the pump?

We are talking about a perfect geostationy orbit system.

So at that point we just keep pumping it in untill it comes out the top. Will that work????

Nope, Nano tubes are up to the task if the wall thickness is enough. If you admit the H2 at the bottom at this point what would happen?

"But you want the whole length of the tube filled with hydrogen or helium, so you can have substantial (multi-kilogram) quantities of gas over a few hours or a day."

Elaborate futher on this? please???

What is the pressure in the tube now that the bottom end is capped?

Ok, cap the end of the tube that is just above sea level. Since the other end of the tube is still in space the whole inside of the tube is now under a vacuum. Correct????? Right????

The gas naturally escaping earths atmosphere is not heated that much nor is it traveling that fast.