The philosophical axiom that
something cannot give what it does not have
nemo dat quod non habet
(cf. I q. 2 a. 3)
applies here.
If the First Mover were movable (i.e., not pure actuality, actus purus), it would lack something (for example:* the ability to move other things), and it could not give what it lacks to others (in the example: it could not change others). This is a contradiction (in the example: because we do observe that things change). Thus, the First Mover must be immovable.
*Any (absolute, not relative) perfection could be given as an example of what the First Mover could lack, and the argument would remain. Other examples: immateriality, omniscience, justice, omnipresence, etc.:
If the First Mover lacked immateriality, it could not give immateriality to creatures; but, we know that there are immaterial creatures; ergo….
Or: If the First Mover lacked all knowledge, it could not give knowledge to creatures; but, we know that there are creatures with knowledge; ergo….
Or: If the First Mover lacked justice, it could not give justice to creatures; but, we know that there are creatures with the virtue of justice; ergo….
Or: If the First Mover were localized to one place, it could not make creatures localized to no place; but, we know that there are creatures not localized to one place (e.g., purely intellectual beings, the angels); ergo….
Etc.