Sorry I didn't see this sooner!

From the docs:
The instanced geometry object, also called the source geometry, can be:

* A single object, animated or not animated.
etc...

So, with that in mind, you can animate your original objects slowly scaling up the way you want, then, using the instancer, tell each emitted particle to start at the beginning of the scale animation.

Then, if you want, you can animate each particle on a Per-Particle basis to fade away after a set amount of time via a ramp mapped to the opacityPP slot, or even create an opacityPP node, and control it via some mel expression, etc.

Good luck!