Question

How do we know that the particles of matter are moving?
In the ink experiment it could have been that the ink flowed due to gravity through the intermolecular space?

Answer 1

Particles are always moving. They have a tendency to move about freely in any direction. But you may not see any noticeable change in the position of your table or your book. Where as as you said the ink got evenly distributed through out water in a little time. Take an icense stick. It will spread its beautiful smell when lit within seconds. This shows us that gas particles move very quickly liquids moderately and Solids doesn't seem to move at all.. In fact Solids do have a tendency to move but unlike liquid and gas solid particles are held tightly together by strong forces that prevent their movement.

Now coming to your second doubt, if the ink particles escaped into the intermolecular gaps then the color of ink would not be visible to us. Sugar dissolves in water because sugar hides itself in the intermolecular spaces which is why once dissolved it becomes invisible. So as the ink is visible it's next to certain that ink particles were moving and hence particles move