This issue is analyzed by Furley in Self-Movers. Aristotle's answer is somewhat convoluted due to conflicting motives, one from common sense and the other from his cosmology. For falling bodies we have the distinction between forced and natural motions, see Rovelli, Aristotle's Physics: a Physicist's Look, and falling bodies move "naturally" towards the center of the Earth without being actively pushed. Yet, as Furley explains:
bodies, as opposed to things with souls, have a source not of causing
movement or of acting but of being acted on. In
fact, this gives too little to the natural bodies in Aristotle's theory. He
should at least stress that they have an internal source of being acted on in a
fully determinate way. But we do not need to pursue that subject here, and
we can also leave aside the difficult question of what is the active mover of
the natural bodies when they move according to their nature - a question
to which Aristotle offers no wholly satisfactory answer."
What "acts on" the "internal source" is presumably their "nature", which draws them to their "natural place". How it connects to the unmoved mover is obscure.
As for the ensouled (animals and humans), on the one hand, Aristotle wants to say that their souls initiate motion. On the other, that would break his cosmological chain that leads to the unmoved mover:
"The tension in Aristotle's thinking about this subject is set up by a clash
of motives. He clearly wants to preserve the commonsense intuition that
the movements of animals, and especially the actions of human beings, are
not brought about by external agents in the same way that the movements
of inanimate beings are. Yet he sees a danger that all the movements in the
cosmos might be thought explicable on this principle of the self-movement
of autonomous parts, and so insists that even this self-movement presupposes
some external changes that are independent of animal movements."
What Aristotle settled on is clever. He says that although self-movers are not moved entirely by themselves, they do, nonetheless, mediate motion in ways qualitatively different from getting pushed by external force or drawn to the natural place, and put enough "self" into it to deserve the name. The "external changes" (aside from metabolism and mechanical stimuli) are objects of desire that "move the soul". Yet they do not move it in a mechanical way of a cart dragging a dog (to use Chrysippus's metaphor), that, in humans, would amount to abolition of moral responsibility.
"Animals are clearly distinguished from inanimate
natural bodies in that although both require external things to explain their
movements, only animals require external things perceived (or otherwise
apprehended) as having significance for them. Note that this is not just a
difference in the complexity of the response to a stimulus, but a difference
in kind. Only a being with a soul can move in this way. An animal is
correctly described as a self-mover, because when it moves, its soul moves
its body, and the external cause of its motion is a cause of motion only because it is "seen" as such by a faculty of the soul."
This is Aristotle's way to have his cake (the unmoved mover as the ultimate source of all motion) and eat it too (get ensouled self-movers to initiate motion autonomously). Does he thread the needle? Perhaps, for his own purpose of getting to the unmoved mover, but not for the later uses in the cosmological argument and/or determinism with sufficient causes along the way leading to the all-determining first cause. On Aristotle's view, soul-mediated effects are not determined by their causes, and so leave room for indeterminism and "free will". In contrast to the early determinist Chrysippus, Aristotle was an early libertarian. His unmoved mover is only the first cause in a weak sense, and his God is no Providence of monotheism:
"The point is that external
objects are not in themselves sufficient causes for the voluntary movements
of animals. But they do have some effect on the soul, and it would be
obstinate of Aristotle to deny that the effect can be called a movement.
There is one conspicuous loose end in the theory that the arche of human
actions is "in" the agent. Aristotle maintains that people are moved to act
by what appears desirable to them, that what appears desirable depends on
their character, and that their character in turn depends on their actions
and is therefore "in their power." His theory needs some explanation of
these character - forming actions and of how it is that they are not caused by
external pressures but proceed from an arche in the agent himself."