Why does hand sanitizer turn into a liquid after coming out of the bottle?
A little research and perhaps a small experiment might help you figure something like this out on your own. First, have you tried wikipedia? A small mention of research can go a long way. Describing what product you are interested in never hurts either. Is the polyacrylic acid, a dispersant in this substance? That material can absorb water effectively, so here is what I would do:

Take a very small amount of water and smear it on the same surface that it does not "turn into a free flowing liquid" and see if it does with the addition of water. If it works, you might conclude that your alcohol-based rubs use a suspension material that when contacted with a little water, such as from your palms, will cause the gel to have just enough water absorbed inside to decrease its viscosity.

Perhaps you will like to share the results of your experiment.

EDIT: OP seems to be observing a gel demonstrating Thixotropy, this I have determined from a quick google search