This is quite possible I think.
In normal chess, black is able to counter white's threats by means of prophylaxis. However in this variant, white can have N+1 threats while black can only make N prophylactic moves.
Let's just focus on the queen. In each turn, she can take N opponent's pieces. Black only has N-1 moves. At some point it is impossible to cover all ways which a queen can use to "dig" the way towards black king.
More formal analysis is of course required to prove the hypothesis but it seems unlikely for this game to be drawish.
This is the variant where White makes 1 move, Black makes 2 moves, White makes 3 moves, Black makes 4 moves. [and a check ends a series]
Other than above, just like regular chess.
My analysis says it seems probable that White has a forced win.
Comments?