If you get caught in a genjutsu that wouldn't stop the puppets from moving if they are in the middle of a jutsu
"Otsutsuki shinobi are capable of embedding their chakra into their puppets to allow them to act on their own recognizance as pseudo-sentient beings, independent of the user's direct control. Because they are free enough to act independently they simulate robots or clones in this regard."
Judging from that line then yeah they could move devoid of your will.
And it would just depend on how you went about doing that but, from your puppet jutsu alone you can't but who knows you could possible create something to do that.
Most basic puppet technique tell me what you think, pros, cons, and just overall explain how it works.
(Kugutsu No Jutsu) - Puppetry Technique
Rank: D
Type: Attack/Defense/Supplementary
Range: Short-Mid
Chakra Cost: 10(-4 each turn its used after wards)
Damage Points: N/A
Description: The user makes threads of chakra that are used to control puppets.
Below jutsu can be used by all puppets so explain to me how they work, include pros, cons ect
(Chakra No Tate) - Chakra Shield
Rank: C
Type: Defensive
Range: N/A
Chakra Cost: 15
Damage Points: N/A
Description: The puppets arm will pop out in four different directions in almost an umbrella-like manner. The arm segments will surround the puppets hand, and from those segments, chakra will emit, each one making one fourth of an entire circle, making a shield of pure chakra.
(Senbon Ame) - Endless Rain of Senbon
Rank: C
Type: Offensive
Range: Short-Long
Chakra Cost: 15
Damage Points: 30
Description: The puppet will open its mouth wide, unleashing a torrent of senbon. All of the senbon are poison-tipped, meaning that a single hit by one could be deadly. These senbon can shower the opponent for as long as the user wishes.