Situation: I am Lillia, on dusk lab, with smoking mirror hidden and 1 sprite in base. Can I swap Lillia with the sprite in base, Kill that sprite with dusk lab, get a draw and, in the end, holding with the "new sprite created by Lillia movement trigger" ?

No — you can get the swap and the hold, but you can't also get Dusk Rose Lab's kill-and-draw in that line.

Dusk Rose Lab's kill is a cost paid at finalization of its trigger, and Beginning-Phase triggers finalize at the start of your Beginning Phase before anyone gets priority to play a Reaction (FAQ #12593; cost-at-finalization under Rule 383.3.b.1). At that moment Lillia is your only unit at the Lab — the Sprite is at your Base — so choosing the Lab means killing Lillia, which kills the swap before it happens. If you decline the Lab, the draw is gone; you can't come back to it later.

What actually works: decline the Lab trigger, then react to the Temporary trigger with the hidden Smoke and Mirrors (Lillia at the Lab + Sprite at Base satisfies both the "different location" clause and the hidden targeting exception, Rule 811.1.d.2.a). Resolve: Lillia → Base, Sprite → Lab, Lillia's move trigger plays a fresh 3-Might Temporary Sprite at the Lab, and Smoke and Mirrors draws you 1. The Temporary trigger then kills the swapped-in Sprite, leaving the new Sprite holding the Lab at the Scoring Step. So you hold and score, but your only draw is Smoke and Mirrors' own 1 — not the Lab's.