simple grid operation

Hey guys,

This might be a simple operation but I have been struggling over it for a while.

As shown in the attached image I have points within a rectangular grid. Every rectangle within the grid is a separate curve. I am trying to project the points onto the rectangles in order to densify it as shown in the image..

I already tried with the Curve CP function to get the projected point on the curve and then join the new and old point with a line, but this only gives me the shortest distance of course.  

Any ideas on how I can do what is shown on the image