I do not understand this problem. Is it not exactly what is supposed to happen? See, for example, attached picture - suppose you move the vertex of left-bottom face - why would there be any reason for texture in top-right face to change? So, naturally there is texture break at the edge after vertex is moved.
The simple way to "fix" this is to split the face in the net of small faces (increase cube subdivisions) and move all of them in proportion. This can be done with as3dmod, for example. Alternatively, you can write clever material utilizing t-s in uvt data to change distortion, or pixel bender shader (neither is easy).
<<attachment: trapezoid.gif>>
