Here is a model to think about. Tweak as needed. I am assuming for sake of my understanding that there is some symmetry about the $xy$ plane, namely there are surfaces $f_1$ and $f_2$ with $0 < f_1 < f_2$ at almost all points in the $xy$ plane (more formally, the numerical relation $0 < f_1(x,y) < f_2(x,y)$ holds for all $(x,y)$ in an open set in the $xy$ plane) and there are also surfaces $-f_1$ and $-f_2$ and what is desired is two surfaces $f_3$ and $f_4$ that satisfy among other conditions $f_1 < f_3 < f_2$ and $-f_1 < f_4 < 0$.
Try a straw model where the volume between $f_1$ and $-f_1$ is made up of parallel straws packed together in the desired direction $u$. If you want volume preservation, push on the straws in the direction u; push so that neighboring straws move close to the same amount. Mathematically, this is reparameterizing the surfaces involved so that the $z$ axis is replaced by the $u$ axis, and then adding a distortion $d$ to the reoriented $f_1$ and $-f_1$.
For an area preserving distortion, try something similar, except shrink or stretch the straws as needed. Instead of $d$ added to both reparameterized surfaces $f_1$ and $-f_1$, you will need to compute the difference in surface areas between $f_1$ and $d+f_1$, and borrow that difference from $-f_1$ somehow. As a start, if an area element gets increased by $b$%, find $c$ to shrink the area element on the other end of the straw so that the net change in the sum of the two areas is zero.
Another model is the soap film model, where $f_1$ is like a soap film with a variety of pressures acting on it. However, this leads to minimal surfaces and/or odd metrics, and is way out of my comfort zone. Perhaps a differential geometer can tell you what an appropriate transformation would be for this kind of model.
Gerhard "Ask Me About System Design" Paseman, 2011.08.22