Of course the distant stars don't "move" at all on these time scales. The Earth moves.... in such a way that all the stars appear to rotate around Earth's poles (and pole stars if there are two at the time) in one day (approximately 24 hrs.). And Earth's pole rotates around the ecliptic pole (at an oblique angle equal to the latitude of the Tropic lines) in one Great Year (approximately 26 thousand years). Most people think of star "movement" in terms of changes in two spherical coordinates, declination (north-south location relative to Earth pole), and right ascension (ra= coordinate relative to the equinox point).
So, any given star travels in a wave-like path (in spherical geometry) in declination and ra coordinates, over a Great Year. Like any two-dimensional harmonic motion, the two different dimensions change at different rates depending on where in the cycle you are. Declination cycles up and down, while ra marches along at the general precession rate (which also gradually varies and cycles but yields on average approximately a 26,000 year cycle). When a star is near north-south culmination (on the Great Year cycle), its declination changes slowly. When a star is midway between culminations, its declination changes fast. Also the shape and amplitude of the wave (and thus precise amount of spherical arc-angle it appears to move per year) depends on the particular star's ecliptic coordinates.
So, to your question "whether or not stars on the ecliptic have a different rate of movement from pole stars" -- the answer is "yes" but you have to be very careful defining what you mean by "rate of movement".