Need help with making a gradual colour change across multiple elements

You can make a gradient along a path in Illustrator. If you start with your white path on the bottom layer, duplicate the layer and change the stroke to gradient along a path and create your color gradient, then duplicate the layer again and change the gradient along a path from black to white you can import the whole thing into After Effects as a composition then set the black to white layer as the source for a Gradient Wipe applied to the colored gradient layer and animate the reveal:

If you carefully examine the screenshots you'll see everything that I did to reveal the gold to red gradient path on layer 2.

This is an easier way to get a soft edge reveal than using trim paths.  If you want to stick with trim paths you'll have to add Trim Paths to a wider copy of the line you want to reveal above it, add a blur, then use that copy as a track matte for the layer below.

The key to making this kind of a transition easy to animate in Illustrator is creating the proper layered illustration in AI before you even start anything in AE.

The "best" way would of course be to split up everything into single elements with individual gradients and animate those, but a simple, reasonably efficient method might be to use the graphics as a track matte on a gradient layer. You're still going to need a few duplicates for the separate blocks, though.

Mylenium