An essential feature of chess is that players are not allowed to "pass". Expiration of a time clock should not affect that.
I would suggest instead that if at any time White notices that Black's clock has fallen and White's clock has not (apply similar rules for the reverse situation), White may punch Black's clock (if White's clock isn't already running) and move for Black, then make his own response, and reset Black's clock to have a some small amount of time left. The ability to compel a Black to make even one a deliberately bad move would usually be a winning advantage, but if Black is otherwise in a vastly better position, Black might prevail even after the penalty is assessed. If at any time both flags are down, the game will be drawn. Further, if Black's flag is down and White wants a draw, White may claim one without having to wait for Black to move and then for his own flag to fall.
The present clock rules would be essentially equivalent to the above but with a zero-second reset (so Black's flag would remain fallen after each take-over move by White). The one difference is that White is awarded a win if there is any sequence of legal moves by which Black could be checkmated, whether or not any sequence of moves checkmating Black could legally be carried out without the game being drawn because of a move limit or (much more rarely) a repeated position.
In case the present board position would not offer White any particular useful opportunities for a take-over move, White might have the further option of setting Black's clock to a small amount of time and asking Black to move immediately. If Black does so, White would retain the ability to make a later move for Black at a time of his choosing; if Black fails to do so, White could force Black's present move and a later one. The only time White could "store up" the ability to make two moves for Black would be on a turn where Black had been told to hurry up and failed to do so. If this rule exists, it would likely rarely need to be carried out (beyond White asking Black to hurry up) but in its absence situations could arise where e.g. Black's flag is down but there are no legal bad moves for Black. In such cases, Black should not be able to force White to spend his take-over move by simply sitting on his hands until White does so.