Can Sketch copy moves that was used on the previous Pokémon, and does this differ over generations?

In Gen 2, if the enemy Pokémon takes you out with a move, and you send out a Smeargle to Sketch it, i believe Sketch will fail? So Sketch has to be used by the same Pokémon the desired move was used on?

In Gen 3 this doesn’t matter? As long as it was the last move used by the opponent? So let’s say you’re taken out by a Thunderbolt, can you then send out a Smeargle to Sketch it?

What if the opponent is inactive due to sleep or paralysis? Can you still sketch the last move they used before that?