Chromatography: Why does the insoluble compound get pulled along?
Chromatography only works when the affinity for the paper vs. affinity for the water is in a dynamic equilibrium.  Suppose there are three compounds in the ink, called $\ce{I}$, $\ce{N}$, and $\ce{K}$.  Lets call the  spots on the paper fiber that bind ink molecules as $\ce{P}$.  Then let's write the dyanmic equilibria this way:

$$\ce{I(aq) + P &lt;=&gt;&gt; PI} $$
$$\ce{N(aq) + P &lt;=&gt; PN} $$
$$\ce{K(aq) + P &lt;&lt;=&gt; PK} $$

This means that any given $\ce{I}$ molecule will be absorbing to paper, then becoming unstuck from the paper, then absorbing to a (probably different) site on the paper, then desorbing again and returning to aqueous solution.  The same is true of $\ce{N}$ and $\ce{K}$.

There are two further facts that we need to realize to understand the chromatography.  The first is that the equilibria are balanced at different points.  This is why the size of the arrows in the reactions I drew above are different.  As drawn, component $\ce{I}$ has stronger affinity for the paper and a lesser affinity for water, as indicated by the longer rightward top arrow and the shorter bottom left arrow.  The reverse is true for component $\ce{K}$, and $\ce{N}$ is somewhere in between.

The second fact is that the water is moving due to the capillarity you mentioned. So when the compounds are in aqueous solution, they will flow with the water.  When they are stuck to the paper, they will not move.  Since all of the molecules are very rapidly transitioning between paper-bound and water-"bound" states, they will all move some of the time, and be still another part of the time.  The key is that the different components spend, on average, different fractions of time moving vs. being still.  This makes their average speed different.

As for the question of why they eventually come to rest, this is because the capillarity-driven flow eventually runs out of steam (i.e. gravitational forces compete with the capillary forces after the fluid rises far enough), and the water itself stops flowing very much.  Thus the bands of separated molecules eventually settle into a relatively fixed position.

So to answer your title question, truly insoluble compounds would not be separated by chromatography.  Compounds need to be soluble to at least a degree in order to be separated.