When there are multiple parallel time-lines, you have to input a move for each one before you can advance to the next turn. But to win, you have to mate just one of the opponent's kings. One player can have multiple kings on one board, because they can move through time and between time-lines.
I have no idea what high-level play will end up looking like, but the streamer from whom I learned the above [0] said that the most common way to win is to mate a king on a board in the past. Those already have their options of escaping an attack restricted by ordinary chess rules; it gets worse when you can make additional threats from the future.