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This question is specifically for felt tip pens and marker pens that contain a felt inner tube. The parts of the pen are: - the body case - the felt nib - the felt inner tube

I do not understand is why the ink within the felt inner tube moves up into the nib and then onto the surface in contact with the nib. Why does the ink not stay in the inner tube?

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Fred Johnson
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In short, capillary action. The fibrous content of the tube is literally designed to be saturated by the nib dries out creating an imbalance in fluid pressure leading to ink flow towards that direction. It also has a lot to do with surface tension in the fluid and viscosity. The more dense the fibrous material, the more viscous the fluid needs to be which is why volatile organic compound are often added to the ink to make it flow better or be more "wet".

Rhodie
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the fibers forming the pen filler are less densely packed than the fibers forming the nib, which extends into the filler a significant distance. This creates a capillarity gradient wherein the nib has a greater affinity for the ink than the filler does, and tends to saturate itself at the expense of the filler. The capillarity gradient guarantees that 1) almost all of the ink in the filler gets used and 2) that the nib doesn't go dry before the ink in the filler us used up.

niels nielsen
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