Basic Color

Go to Particle Update and add a Color from Curve module. This will allow you to generate colors based on a gradient that you determine in the emitter using Particles.NormalizedAge as the default dependent variable.

Beginning of life has a value at 0 on the axis, and at the end of the life has a value at 1 on the axis.

Advance Color Based on Distance

Below replaces the Particles.NormalizedAge with a Normalize Distance, using the particles position as the start and the Simulation position as the end. A distance of 400 is where the start begins to take effect.