Does Hextech Ray miss if the opponent moves the targeted unit to the other battlefield?

No — it still hits. Hextech Ray only requires its target to be "a unit at a battlefield," not a unit at a specific battlefield, so moving that unit to another battlefield leaves it a legal target and it still takes the 3.

Compare the location-locked cases: Elder Dragon's play effect (FAQ #13023) and Void Seeker both mistarget when the unit moves, because their instructions care about which location the unit was chosen at. Hextech Ray's instruction has no such anchor — only a unit that leaves the board entirely or moves to Base (no longer "at a battlefield") becomes illegal and gets skipped (rule 359.3.e.5).

Note that the damage happens when the spell resolves; if the move was made in reaction on the chain, they still get hit wherever they land on a battlefield.