If we take Aquinas' first way, for example, the inference that a chain of movers exist, is readily made, but no defense for this assumption is given.
He proves "that everything that is moved is moved by another" (quidquid movetur ab alio movetur) and "that in movers and things moved one cannot proceed to infinity" more at length in Summa Contra Gentiles I cap. 13 than he does in Summa Theologica I q. 2 a. 3.
Here are his proofs of quidquid movetur ab alio movetur from Summa Contra Gentiles I cap. 13 [5]-[10]:
[5] The first of these propositions [quidquid movetur ab alio movetur] Aristotle proves in three ways. The first way is as follows. If something moves itself, it must have within itself the principle of its own motion; otherwise, it is clearly moved by another. Furthermore, it must be primarily moved. This means that it must be moved by reason of itself, and not by reason of a part of itself, as happens when an animal is moved by the motion of its foot. For, in this sense, a whole would not be moved by itself, but a part, and one part would be moved by another. It is also necessary that a self-moving being be divisible and have parts, since, as it is proved in the Physics [VI, 4], whatever is moved is divisible.
[6] On the basis of these suppositions Aristotle argues as follows. That which is held to be moved by itself is primarily moved. Hence, when one of its parts is at rest, the whole is then at rest. For if, while one part was at rest, another part in it were moved, then the whole itself would not be primarily moved; it would be that part in it which is moved while another part is at rest. But nothing that is at rest because something else is at rest is moved by itself; for that being whose rest follows upon the rest of another must have its motion follow upon the motion of another. It is thus not moved by itself. Therefore, that which was posited as being moved by itself is not moved by itself. Consequently, everything that is moved must be moved by another.
[7] Nor is it an objection to this argument if one might say that, when something is held to move itself, a part of it cannot be at rest; or, again, if one might say that a part is not subject to rest or motion except accidentally, which is the unfounded argument of Avicenna. For, indeed, the force of Aristotle’s argument lies in this: if something moves itself primarily and through itself, rather than through its parts, that it is moved cannot depend on another. But the moving of the divisible itself, like its being, depends on its parts; it cannot therefore move itself primarily and through itself. Hence, for the truth of the inferred conclusion it is not necessary to assume as an absolute truth that a part of a being moving itself is at rest. What must rather be true is this conditional proposition: if the part were at rest, the whole would be at rest. Now, this proposition would be true even though its antecedent be impossible. In the same way, the following conditional proposition is true: if man is an ass, he is irrational.
[8] In the second way, Aristotle proves the proposition by induction [Physics VIII, 4]. Whatever is moved by accident is not moved by itself, since it is moved upon the motion of another. So, too, as is evident, what is moved by violence is not moved by itself. Nor are those beings moved by themselves that are moved by their nature as being moved from within; such is the case with animals, which evidently are moved by the soul. Nor, again, is this true of those beings, such as heavy and light bodies, which are moved through nature. For such beings are moved by the generating cause and the cause removing impediments. Now, whatever is moved is moved through itself or by accident. If it is moved through itself, then it is moved either violently or by nature; if by nature, then either through itself, as the animal, or not through itself, as heavy and light bodies. Therefore, everything that is moved is moved by another.
[9] In the third way, Aristotle proves the proposition as follows [VIII, 5]. The same thing cannot be at once in act and in potency with respect to the same thing. But everything that is moved is, as such, in potency. For motion is the act of something that is in potency inasmuch as it is in potency. That which moves, however, is as such in act, for nothing acts except according as it is in act. Therefore, with respect to the same motion, nothing is both mover and moved. Thus, nothing moves itself.
[10] It is to be noted, however, that Plato, who held that every mover is moved [Phaedrus], understood the name motion in a wider sense than did Aristotle. For Aristotle understood motion strictly, according as it is the act of what exists in potency inasmuch as it is such. So understood, motion belongs only to divisible bodies, as it is proved in the Physics [VI, 4]. According to Plato, however, that which moves itself is not a body. Plato understood by motion any given operation, so that to understand and to judge are a kind of motion. Aristotle likewise touches upon this manner of speaking in the De anima [III, 7]. Plato accordingly said that the first mover moves himself because he knows himself and wills or loves himself. In a way, this is not opposed to the reasons of Aristotle. There is no difference between reaching a first being that moves himself, as understood by Plato, and reaching a first being that is absolutely unmoved, as understood by Aristotle.
Also, from here, see these articles related to quidquid movetur ab alio movetur:
The Law of Inertia and the Principle Quidquid Movetur ab Alio Movetur by Antonio Moreno, O.P.
α. The Aristotelian definition of motion and the principle of inertia, a thesis by McLaughlin, Thomas J., University of St. Thomas (Houston), author of "Nature and Inertia" in The Review of Metaphysics, Vol. 62, No. 2 (Dec., 2008), pp. 251-284
cf. these other articles by McLaughlin, who is an expert on Aristotelian motion and the principle of inertia.
Usually the argument is interpreted so, that all the actualizations of the potentialities happen instantaneously. There exists no temporal ordering between the different movements of the corresponding movers. But how can we be so sure that no such temporal ordering exists?
As mentioned here, in Summa Theologica I q. 46 a. 2 ad 7,
St. Thomas distinguishes between causal series ordered per accidens ("accidentally") and causal series ordered per se ("essentially"). Edward Feser's Aquinas: A Beginner's Guide (ch. 3 "Natural Theology," § "The First Way") describes the difference.
The full quote from Feser is here. Feser says "Causal series ordered per accidens are linear in character and extend through time" and "Causal series ordered per se are paradigmatically hierarchical with their members acting simultaneously". It seems you only think St. Thomas is speaking of "Causal series ordered per se."
See also William Wallace's "Aquinas on the Temporal Relation Between Cause and Effect."
And if it does exist, what justification can we give to reject such loops of movements? Why can't a mover induce a change in the “moved”, which in turn again leads to the “moved” inducing a change in the original mover, and assume such a loop goes on ad infinitum?
In other words: "Why can't 'quidquid movetur ab alio movetur' be false?"
