The following is not an exposition or explanation of any argument found in the texts of Aristotle or Thomas Aquinas, but it is inspired by their work, and located within the tradition.

​Let's begin by assuming that all change must have a cause. Another word for ‘change’ in this context is ‘passion’. Let’s assume, then, that every passion has a corresponding action. On this picture, the action of the agent is the cause; the passion of the patient is the effect. Causation always involves two or more substances. Here we will also introduce a principle of proportionate causality. It is obvious, for example, that actual change or passion can only result from actual action, involving an actually existing agent and an actually possessed active power. A merely potential event cannot be the actual cause of any actual change. A merely potential agent cannot act.
 
If we were to abandon this principle of ontologically proportionate causality, we would have no explanation for the asymmetry and irreflexivity of causation. That is, we couldn’t explain why a given passion couldn’t be its own cause, promoting itself from mere potentiality to actuality. This would be tantamount to rejecting the causal principle altogether.
 
Every passion must be located in time, since time is the measure of change. What is the temporal relation (if any) between an action and its corresponding passion? There are four logical possibilities: (i) the action is earlier than the passion, (ii) the action and passion are simultaneous, (iii) the action is later than the passion, or (iv) the action is unlocated in time. I will argue that only cases (ii) and (iv) are metaphysically possible.
 
Let’s say that an entity is temporal when it has a state that is located in time. In cases (i) through (iii), the agent has a state (namely, the action) that is located in time, so the agent must be temporal. Only in case (iv) can we have an atemporal or timeless agent.
 
If an agent is temporal, then all its states are actual or potential only relative to the various moments of time (see Koons 2020, Koons forthcoming). Therefore, we cannot say that the agent’s action is actual simpliciter but only that it is actual or potential at this or that time. We must also adapt our principle of causality to incorporate this relativity: for each passion, its corresponding action must be actual at the time at which the passion occurs. Actions occurring in the past or future are, at the time of the passion, merely potential. Hence, we can rule out cases (i) and (iii).
 
Every change must have a cause. If a temporal agent A acts at time t to produce a passion in some patient, then agent A must have undergone some change that eventuated in this particular action at time t. The agent has changed from not being the agent of a particular change to being the actual agent of that change. Hence, the change in the state of the agent requires a cause.
 
If all agents were temporal, this would lead to at least one infinite causal regress at each moment of time. We could then consider the whole plurality of things undergoing change at that time and ask, What causes them to change? Since these changes are all simultaneous, nothing prevents us from aggregating them together into a single, massive event. Given the principle of causation, this simultaneous plurality of events must have a cause that is both separate from itself and actual (at t). Since the plurality includes all changes occurring at this time t, the only possible cause of the plurality of changes would be the action of an atemporal agent. An atemporal action can act at any or all times without undergoing any change itself, and so without requiring a cause.
 

Thomas Aquinas, building on arguments found in Books 7 and 8 of Aristotle's Physics, argues in The First Way (in Summa Theologiae I q2 a3, and in a parallel passage in Summa Contra Gentiles I.13) for the existence of an unmoved mover. However, there is an evident gap between such an unmoved mover and God. At the very least, Aquinas needs to show that the unmoved mover is absolutely unmovable (in all respects). If he can establish that, then he can conclude that the First Mover must exist outside of time. And, in order to exist outside of time, the First Mover must lack all passive potentiality (i.e., be a being of Pure Act).
 
Aristotle and Aquinas are both well aware of this gap, and they have a definite strategy for filling it. The argument goes something like this:
1. Assume (for contradiction) that the First Mover is changeable in some respect.
2. Necessarily, time passes if and only if change occurs.
3. All motion in fact depends on the activity of the First Mover (established by the main argument of the First Way).
4. If the First Mover were changeable in any respect, then it could be in a state in which it failed to cause any motion.
5. To be in such a state, the First Mover would have to be in that state for some period of time (since nothing can be in a state in a single instant).
6. Since all motion in fact depends on the activity of the First Mover, if the First Mover were in a state in which it failed to cause any motion, there would be no change during the period in which it is in that state.
7. If there were no change during that period, time would not pass during that period.
8. If a period has a temporal duration, time must pass during it.
9. There would be a possible state of the world during which time both does and does not pass. Contradiction.
10. So, the First Mover cannot change in any respect.
 
The crucial premises are 4, 5, and 6. Let me take 5 and 6 first.
 
Premise 5. This is based on Aristotle's resolution of Zeno's paradoxes in the Physics. Instants of time are not parts of time--they are only boundaries of such parts. Hence, nothing happens during an instant. Nothing can be in a state of activity or inactivity for only an instant. Instants can only mark the beginning or end of a period of activity or inactivity.
 
Premise 6. This depends on a kind of subtraction principle. If all change in the actual world depends on the First Mover and there is a possible state of the First Mover in which it would cause no change, then there is a possible world where no change occurs. We can simply subtract the activity of the First Mover from the actual world without being forced to add any new source of motion.
 
So, the crucial assumption is premise 4.  Suppose the First Mover is changeable in some respect. Why think that it must be changeable into a state in which it would cause no motion at all? Why couldn't it have a nature such as to cause motion in every possible internal state, while admitting of more than one such possible state?
 
There is some plausibility to the idea that the activity of a thing must depend on the thing's internal state, and that this dependence entails that there be some internal state in which no activity would result. However, this seems far from airtight to me.
 
I think there's a better strategy for defending premise 4--one that is not explicit in the texts of either Aristotle and Aquinas, but which lies quite close to their conception of time and motion.  If something is changeable in any respect, then it lies within time. If a thing lies within time, then its natural activity through time depends on the metaphysically prior passage of time. So, it is impossible for the activity of something changeable to be the ground for the passage of time itself. Yet, that is exactly what the First Mover must do. Hence, the First Mover must be absolutely unchangeable.