We can detect the cognate "to move" in the verb muovere. In English, "to move" can be either transitive or intransitive.
We can move a piece of furniture from one place to another, or we can be the ones to move on our own. In Italian, however, muovere is basically transitive, in its natural, non-reflexive form.
Per me la cosa più bella è recitare e muovere i pupi.
For me the best thing is reciting and moving the marionettes.
Caption 56, Dottor Pitrè e le sue storie - Part 11
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Non riesco a muovere la gamba (I can't move my leg)!
When it's intransitive, it is primarily used in its reflexive form.
Il nostro uomo sta per muoversi.
Our man is about to move.
Caption 23, Provaci ancora prof! S1E3 - Una piccola bestia ferita - Part 23
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In addition to merely moving around in space, muoversi is used a lot to mean "to get going," "to get moving" (also figuratively), or "to get some exercise."
Ti vuoi muovere? -Arrivo!
You want to get moving? -I'm coming!
Caption 25, Provaci ancora prof! S1E1 - Il regalo di Babbo Natale - Part 11
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Non ti muovere o sparo!
Don't move or I'll shoot!
Caption 28, Un Figlio a tutti i costi film - Part 6
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Lui ha detto: "Io da qui non mi muovo."
He said, "I am not budging from here."
Caption 10, Fuori era primavera Viaggio nell'Italia del lockdown - Part 14
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The verb muovere has an irregular conjugation, and the past participle is used quite often as an adjective.
When the sea is rough, it's il mare mosso.
Non lo vedo più. -Perché il mare è un po' mosso.
I can't see him anymore. -Because the sea is a bit rough.
Caption 50, PIMPA S3 EP12 L'amica Onda
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When your hair is a bit wavy or not combed neatly, we use the past participle mossi. Let's remember that, in Italian, we use the plural capelli, even though in English, hair is a collective noun.
Aveva dei capelli mossi (she/he had wavy hair).
When you want someone to hurry up, you can say, muoviti (hurry up, get moving)!
Oh, cammina, muoviti. -Aspetta.
Hey, get going, move it. -Wait up.
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There is a noun form that comes from the past participle, and that is la mossa (the move). When you make the right move, fai la mossa giusta.
When you need to get a move on, it's darsi una mossa (literally, to give oneself a move).
Allora ragazzi, bisogna che ci diamo una mossa.
So, guys, we need to get a move on.
Caption 20, Concorso internazionale di cortometraggio A corto di idee - Part 1
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Datti una mossa, dai (get a move on, come on)!
Of course in English, we use the verb "to move" when we go to live in a different apartment or house. You may be wondering how to say that in Italian. Transitive or intransitive? None of the above!
The verb is traslocare, or, much more common, fare trasloco. Think of it as "translocation!" or "translocate," a cross between "transfer" and "relocate."