What if the 2nd move was 2. Kc5 Kc7? Why would black move purposely so he can lose the match?
True, 2. ... Kc7 delays white's win by three moves. White has a faster win (correct me if I'm wrong) with:
Oh, and to the original question. Can't think of a meaningful puzzle like that, except white (black) to move and draw. But puzzles with a Rundlauf (round trip) motif (a piece eventually returns to its starting square) are not rare.
I'm interested to know if there is a puzzle with repeating moves. What I mean by this is, white to move and win. White makes a move with a piece. Black makes a move. White moves the piece back to its previous square. Black makes a different move. And so on, where black is making unique moves, with the forcing line for White being to move its own piece back and forth.
I can imagine something where there are pins on both sides, so by moving the piece a different pin is broken each time (maybe for both sides), allowing a different responding move.