Does the Particle Animator work with transparent shaders?

When using alpha-blended shaders (either the ones found in the Particles/ shaders section, or in the Transparent/ section), the Particle Animator doesn't seem to want to animate my particles. I'd like them to fade out smoothly, and if I add a particle animation that fades them out, nothing happens.

Yes, it does.

The one in Particles/Alpha Blended should work as you're expecting. I don't think the regular transparent shaders in the transparent family will though.

I just tried this with the "palmbranch" texture from the standard assets, using the Particles/Alpha Blended shader, and it fades the particles in and out fine, while also preserving the alpha of the texture.

To reproduce my test, start a new project (include standard assets), create a new material, put the "palmbranch" texture on it, and set the shader to "Particles/Alpha Blended". Then create a new particle system, and apply your new material to it.