falling start battlefield to base

No — a unit killed by Falling Star goes to its owner's Trash, not base. Killed units (whether from lethal damage or a Kill effect) are placed in their owner's Trash (Rule 323.5 / 428); only Recalls send units to base (Rule 454).

Also, Falling Star has no location restriction — "Deal 3 to a unit" can hit a unit at base or at any battlefield. (Falling Comet is the spell restricted to "a unit at a battlefield.") If you meant something else by "start battlefield to base" (e.g., moving a unit via Standard Move), clarify and I'll rule on that.