In this example Particles Emitter is small IcoSphere following spiral Curve.
Brownian - will scatter particles.
Damp - will keep them together.
Gravity - well, will make them less influenced by gravity :)
IcoSphere.001 - is our Particle Object.
Try them out to get desired result.
(Constant rewinding video on YouTube is not my strong side.)
Particle Object material:
To the Particle Object (in this case IcoSphere.001) you will want to assign Particle Info node and change color of them by their Age, or read more here: How do I get particles to change color over time?
Math node > Divide allows you to change how quickly this change will be and ColorRamp gives you colors.