Using Jyn and her nimble ability, does she get to move out of turn or is the movement saved? I just wanted to make sure.

She moves out of turn. Whenever a model gains movement points outside of its activation, it has to spend them immediately.

Jyn is nuts. I used Smuggled Supplies on her one round, granting her an extra evade. Boba Fett attacked her, she rolled an evade, meaning she had a total of two. so she got to move four spaces for free. On top of that, then boba moved adjacent to her, and I threw down Slippery Target, moving another five spaces.

9 Spaces not even on Jyn's activation!