1. Can Aquinas handle cases of motion by nature?
1a. The mover of what is moved by nature is that which makes the motion possible (either the generating cause or the cause that removes impediments to action).
One of the arguments Aquinas discusses in Summa Contra Gentiles 1.13 is the following:
Omne quod movetur per accidens, non movetur a seipso. Movetur enim ad motum alterius. Similiter neque quod movetur per violentiam: ut manifestum est. Neque quae moventur per naturam ut ex se mota, sicut animalia, quae constat ab anima moveri. Nec iterum quae moventur per naturam ut gravia et levia. Quia haec moventur a generante et removente prohibens. Omne autem quod movetur, vel movetur per se, vel per accidens. Et si per se, vel per violentiam, vel per naturam. Et hoc, vel motum ex se, ut animal; vel non motum ex se, ut grave et leve. Ergo omne quod movetur, ab alio movetur.
My very rough translation/paraphrase:
Every that is moved per accidens, is not moved by itself. It is therefore moved to motion by another. Likewise, neither what is moved violently: as is obvious. Neither are those things moved by nature as self-moving, such as living things, which as is well known are moved by the soul. Nor again the things that are moved by nature as heavy and light. For such are moved by what generates and removes impediments. And everything that is moved is moved either per se or per accidens. And if per se, either violently or naturally. And if naturally, either they are moved from themselves, as a living thing; or they are not moved from themselves, as heavy and light. Therefore everything that is moved, is moved from another.
This is what Aquinas calls an 'inductive' argument. For the scholastics, an 'inductive' argument usually involves a division into classes of the entire possible range of actions, followed by the proving or disproving of a claim with regard to each class. This is what Aquinas does here. We need to navigate somewhat carefully, because some of the examples are from Aristotelian physics that is no longer held; but because the division proceeds logically, it actually can stand without the physics. Everything moved is either moved per accidens or per se. Motion per accidens is moved by another, simply as part of the definition. So the question is whether there is any per se motion that is not moved by another. Aquinas divides the field of per se motion into things that are moved violently or naturally. Being violently moved means being moved by something in spite of one's nature; so it is an obvious case in which motion is from another. So we come to things that are moved by nature; and Aquinas divides this field into things that are moved by nature because they are self-movers, and things that are moved by nature that are not self-movers. Aquinas thinks that things that are self-movers as a whole are cases in which we are dealing with (as it were) a system of parts in a mover-moved relation: i.e., every case of self-motion is a system, consisting of parts, within which the principle 'what is moved is moved by another' applies. So that leaves cases that are moved by nature but not as self-movers. In such cases the nature acts on its own; but mover is in such a case the cause that either generates the thing with the nature (causes it to be this sort of thing with this sort of nature) or else (depending on what precisely we are looking at) what removes impediments to the action of the nature.
So Aquinas has no problem with natural motion; in fact, his own view of the universe is filled to the brim with it. But natural motion, of itself, is not a threat to the principle that 'what is moved is moved by another'; it just is a special case of it. But we need to look at another, related objection.
1b. Newtonian mechanics does not rule out Aquinas's principle.
The most common objection to Aquinas's First Way, I think, is the argument from Newton. Clayton Littlejohn gives the basic idea:
If Aquinas is going to get anywhere from the starting point 'Something is moving', he has to introduce a principle to the effect that if this is true, there is something external to the object that accounts for this unless the object is a self-mover. Such a principle conflicts with the principles of Newtonian mechanics which requires the introduction of an external body or force to explain changes from motion to rest or rest to motion but nothing to explain the persistence of a state of rest or motion.
Now, I think the first thing that needs to be recognized is that it's actually difficult to find any clear conflict between Newtonian mechanics and the principle that is being applied. (1) It isn't clear that Aquinas is committed by the principle to saying that objects need something explain their persistence in a state of rest or motion (indeed, what we saw in [1a] suggests otherwise). (2) Even as late as the mid-nineteenth century, we have people like William Whewell who, in interpreting the philosophical implications of Newtonian mechanics (and there are few who do so in such an extensive way as Whewell does), regard the law of inertia as derived from a causal principle (rather similar to Aquinas's principle about motion) combined with restricting suppositions. In other words, an argument can be made, and was made, that the law of inertia is actually just a more restricted version of a causal principle like Aquinas's. (3) Even if we set aside all motion in our modern sense of the term, the Latin motus is a broader term encompassing all changes in which something potentially x becomes actually x; so there would still be the question of whether any of these changes allowed the argument to work (all it needs is one). So unless one allows that no change requires any causal explanation, it really isn't possible to head off the First Way in this way.
So, ultimately, I'd really need to see the analysis according to which Aquinas's principle is said to conflict with Newtonian mechanics; there doesn't seem to be any reason to think there is any actual conflict. (Which is not to say that Aquinas didn't believe some things that would conflict; but the issue is whether the principles actually used in the First Way would conflict.)
The second thing to note is that Aquinas, in allowing that the universe may always have existed (i.e., that the past may be infinite), is necessarily (given an Aristotelian view of the universe) allowing that motion may always have been. So he isn't committed to the view that there was once a time when everything was in stasis. (One can argue whether it actually makes sense for their to have been motion from always; if I recall correctly, Franz Brentano in one of his lectures on the existence of God argues that on such a supposition, motion would necessarily be undefined and indeterminate. But this isn't Aquinas's way.)
2. Aquinas's arguments do not use the claim that "what causes X to be F must itself be F," rather, they use the claim that "what causes X to be F must itself be."
I blame this confusion on Anthony Kenny's The Five Ways, where he translates the premise correctly but consistently interprets incorrectly. When Aquinas uses this premise, the idea is not that what causes X to be F must actually be F, but that what causes X to be F must actually exist. The former would, in fact, contradict a fact about divine causation that Aquinas elsewhere recognizes, and at great length: it is equivocal causation, and it is precisely the nature of equivocal causation that what causes X to be F need not be F at all. The culprit appears to be reading an example as if it were a paradigm case rather than what it really is, an argument. Sentence #4 in my translation reads:
4. For moving is nothing other than drawing forth something from potency into act, for something cannot be reduced from potency into act, save through some actual being [nisi per aliquod ens in actu]; thus actual heat (as fire) makes wood (which is potential fire) to be actually hot, and thereby moves and alters it.
What misleads people is taking the example (the actually hot making wood to be actually hot) as if all cases were like this. But this is not required by the way scholastics use examples. For a scholastic an example is an abbreviated inductive argument; often a case is chosen in which the principle is very easily seen, and then (as it were) carrying out the induction in other cases is left to the reader. Aquinas doesn't use the example of an actually hot cause making something actually hot because all cases are like this, but because cases like this are obvious cases in which the principle applies. And, indeed, any case in which what causes X to be F is actually F will necessarily also be a case in which what causes X to be F is something actual. People like Kenny are reading the particular case as if it were the general principle. Thus, Aquinas does not hold that God has the properties he causes (which would introduce a rather obvious contradiction into all of the Five Ways), but is merely noting the much less controversial claim that things can only be caused to be actually anything by things that are themselves actual. This means that being moved requires some prior actuality, which brings us to the next proposition.
3. Mutual motion, in the sense that would be relevant to the argument, is impossible.
Given that the issue is actuality and potentiality, mutual motion doesn't appear to be a legitimate counter-response; since the only mutual motion that would actually evade the argument is a case in which A under aspect a makes B actual under aspect b in such a way that B under aspect b makes A actual under aspect a. In other words, to go this way would actually require that we deny that there is any such motion, or deny the premise that "What makes something actually F must actually exist"; since by our supposition B-under-aspect-b depends for its existence on the priority of A-under-aspect-a and A-under-aspect-a cannot exist unless B-under-aspect-b is prior. This would imply that B-under-aspect-b is able to cause its cause prior to its actually existing. There are, of course, senses of 'mutual motion' which are perfectly harmless, e.g., A under aspect a1 making B actual under aspect b1, which in turn makes A actual under aspect a2, which in turn makes B actual under aspect b2, etc. But this wouldn't pose any problem to the argument. (It's also worth pointing out that, even on the supposition of mutual motion, this would not necessarily break the argument unless all asymmetric causal dependencies traced back to symmetric ones.)