Which cytoskeleton components move chromosomes during cell division?
it is microtubules.
MTOC (Microtubeles organizing centre) is the one which takes care of oroducing Microtubules. these micryubulrs extend and gets attached to the kinetochore region of chromosome and they move the chromosome either by adding more of the alpha tubulin or removing to lengthen or shorten the microtubules. http://www.nature.com/scitable/topicpage/mitosis-and-cell-division-205 figure 5 and 6 are the real deal for giving you more insights on this.