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Say Player 1 has something like Shimmer Myr that reads:
You may cast artifact cards as though they had flash.
Then when that player tries to cast an artifact as an instant, Player 2 places something like Naturalize on the stack to destroy target artifact (the Shimmer Myr). What happens to the artifact being cast? What about the mana used to cast it?
Player 2 can't really stop player 1 from casting an artifact as if it had flash, outside of destroying Shimmer Myr while player 1 is tapped out.
At any given time, one player has priority. If player 2 has priority first and casts Naturalize, player 1 can respond by casting an artifact. If player 1 has priority, they can cast an artifact. Once they do so, it's on the stack and it doesn't matter if player 2 casts Naturalize; the artifact has already been cast and will eventually resolve unless it's countered.
That is, Shimmer Myr's ability only matters while they're casting the artifact. (Same goes for flash.) You check whether you're allowed to cast a spell as part of casting it, not later, and timing restrictions are part of that.
