The doctrine of immutability, in its weak version, states roughly that God cannot change in character. In contrast, the strong version states that God cannot change at all. Proponents of the strong version have traditionally appealed to either the perfection of God or the argument from motion to the existence of an Unmoved Mover in support of their view. The more I think about this, the more I'm persuaded that defenders of the weak doctrine can also affirm the argument from motion. First, let's think about what the argument entails:
1. Evident to the senses is motion. (Premise)
2. Everything in motion is moved by another. (Premise)
3. Either a first mover exists, or else there is an infinite regress of movers. (Premise)
4. There cannot be an infinite regress of movers. (Premise)
5. Therefore, a first mover exists. (From 3 and 4)
Defenders of the argument are quick to point out that the first mover must itself be unmoved, since otherwise it would be moved by another and wouldn't be first, which is contradictory.
The question is this: why couldn't the proponent of weak immutability affirm this argument and maintain that only a part of God is immutable? This would be especially fitting given that weak immutability is usually affirmed by those who also affirm the doctrine of divine simplicity in its weak form. While God is not composed of composite physical parts, his attributes are still distinct from one another.
This type of motion may be likened to one who is moved by the beauty of a painting. The painting does not have to change in order to move the viewer. (Of course, a painting does move in certain respects, e.g. at a molecular level.) The way this would work, then, is that God's will and his actions are moved by the beauty of his immutable character.
Now, the strong doctrine of simplicity entails strong immutability, so the truth of the former implies the truth of the latter. I won't go into that, though, since we're only talking about the coherence, and not necessarily the actuality, of weak immutability with the argument from motion.