Why does the "gap" between atomic radii get smaller down the periodic table?

The question asks: why is the difference in atomic radii between $\ce{K}$ and $\ce{Na}$ larger than the difference in atomic radii between $\ce{Cs}$ and $\ce{Rb}$? I do have some kind of notion that the gaps between shells decrease as you go down the periodic table, but why would this be the case?