Check out these Indian musicians (the sax playing starts around 13 minutes in, after the drum battle/conversation). There is no movement without music. In some cultures there aren’t separate words for music and dance. Using the body to keep track, to groove, and to make better music is something all musicians do. The hand gestures you see in this video are one of the many ways Indian musicians manifest the music physically.
To me, improvising is like speaking a language spontaneously, whereas only reading music and re-creating the music others (or yourself) have written is like reading a book or a story out loud. The reading is scripted, whereas the spontaneous use of language flows where it will, especially if it’s in conversation with another. If you’re new to improvising, it’s a simple thing to do.