If enemy Armored Tusker moves into a battlefield I control with Leona, Zealot, and I Zenith Blade his unit on another battlefield and move my Leona, Zealot there, does Armored Tusker still do damage to Leona, Zealot?

No — Leona, Zealot still takes the 3 damage. Zenith Blade can't be played in the window you're describing, and even if it could, moving her to another battlefield wouldn't shake the trigger.

Armored Tusker's "When I move" trigger goes onto the chain in the Cleanup right after the move, and no Showdown can open while the chain has items — so it resolves before any Combat Showdown exists. While that item is pending the turn is a Closed State, where only cards with [Reaction] can be played (Rule 309.1.a); Zenith Blade is an [Action] spell, so you have no legal window to cast it in response. It resolves, Tusker's controller picks Leona, Zealot, she takes 3.

Even if you could respond, the trigger's target is locked at finalization and its restriction is just "a unit at a battlefield" — not "here." Moving her from one battlefield to another never stops her being a legal target, so she'd still take the 3 (compare the Elder Dragon ruling, where the restriction was per-location and a move genuinely broke it). To dodge it she'd have to leave the board, and Zenith Blade can only move her to the enemy unit's battlefield.

Your real window is the Combat Showdown afterwards: casting Zenith Blade there to move Leona away does prevent combat damage, but the 3 is already marked — and if she leaves, nothing is defending and the Tusker conquers. Stunning the Tusker itself in that Showdown is the play that actually blanks its combat damage.