This is a weird little game - regular checkmates are often not, because the king can just escape back in time or across timelines, but threatening a king's past

This is a weird little game - regular checkmates are often not, because the king can just escape back in time or across timelines, but threatening a king's past self is pretty easy - it can't escape having been where it was. And you only need one checkmate to win, because you have to make a move on all boards where it's your turn (even if you have no pieces there - you have to move one there!). You can also give check with a king, because of the quirk that only "the present" can threaten check. At least, I think that's why it works.