引っ越しする　（ひっこし する）is the verb specifically used for a change of residence. You can leave out the first "し" to have it flow better. For example, "I moved to Japan" would be 日本に引っ越しました。 (にほん に ひっこ しました。）

depends on what kind of movement you are talking about . Like if you want to say " I can't move my legs" you use the verb 動く。If you move out from your house , you use 引っ越す　か　移る。You can also say 移動する, for example if you are traveling to somewhere. 変わるmeans to change.