Why do they call it a "movement" in music? Is this a metaphor?

I can't understand this. In symphony you hear the word "movement" to refer to independent sections. But it seems that it's not really that relevant. Movement almost everywhere has something to do with motion. Is this nomenclature a kind of metaphor for something in music, like moving from one section to another?

The earliest meaning of movement in music was how it moved: its tempo or relative speed (probably referring to the natural movement associated with the beat of the tempo). For instance, the OED quotes an entry from Baily's 1721 dictionary:

Allegro: A Term in Musick when the Movement is quick.

Similarly, Henry Purcell's Sonnatas of III Parts (1683) in its preface speaks of "Allegro, and vivace, a very brisk, swift, or fast movement."

As musical forms with multiple sections, such as symphonies, evolved, each of the sections generally had a different tempo, or movement (which was a main reason for having distinct sections). One would refer to a section of a work by its tempo, e.g. the allegro movement or the largo movement. A natural evolution was then to speak of the several movements of the work, that is, the several sections of different tempos.

The usage of movement in reference to tempo per se has largely died out, but the reference to movements as sections of different tempo has continued. We still often find a tempo term (e.g. vivace) as the title or part of the title for a movement.

The OED determines two senses in which "movement" is used in music - Senses 4a & 4b.

4a.The manner of transition from note to note or passage to passage in a piece of music; relative speed, tempo. Now rare.

The earliest example is from 1683.

1683 H. Purcell Sonnata's of III Parts Pref. Allegro, and vivace, a very brisk, swift, or fast movement.

Though it is described as "rare" there are examples through to the present time.

4b.A principal division of a longer musical work, usually differing in tempo from the other divisions and having a distinctive character of its own. Also in extended use.

Its earliest mention is from 1694.

1694 Philos. Trans. (Royal Soc.) 18 69 Upon the occasion of the difference of 3ds, and the difference of 6ths, he discourses which of them may be more properly made use of in movements of Consort-Musick.

This question seems to be more about 4b than 4a. However, though the OED doesn't say so, it would seem plausible to me that sense 4b developed out of 4a.

I looked up movement related to dance. That is sense 1b, but interestingly the earliest example post-dates that of music.

1b. Dance. A change of position or posture; a step or figure.

1715 R. Steele Lover 24 Mr. Siris has made the beginning of this Movement very difficult for anyone who has not, from his natural Parts, a more than ordinary Qualification that way.