I dont really believe in pushing as if through keybed - just using as much weight as is necessary to put the key down (pushing to the floor just wastes energy as far as i can see).
You need to learn to push through the keybed. Playing on top of the keys works fine on the piano you are accustomed to playing with, but will completely fall apart on a piano that feels different. pushing through the keybed promotes a good solid foundation and the ability to play no matter how heavy the piano is.