I am not sure what you ultimately want to do with these curves, but the FDE did a nice job with them!
The dynamic time warping aligned all four curves. I aligned them to curve A. A linear P-spline model fits the data well.
The first two functional principal components are probably sufficient to capture the differences between these four curves.
The plot shows the A and B are most alike and most different from D based on the first FPC. C is neutral in this way. A, B, and D are most alike and most different from C based on the second FPC.
This profiler will let you vary the FPC values to see how they capture the differences in the shape. Remember that their effect decreases in magnitude from FPC1 to FPC 3.
You could add supplementary variables, such as factors in a DOE and then link the factor effects to the curve shapes.