Is there a way to stagger start times for replicator cells?

I have keyframed a simple shape to move downwards with custom easing, so I can't use the replicator behaviour for this, as that only lets me keyframe the movement of every replicated shape as a whole.

If an emitter is not an appropriate alternative:

If you place the animated shapes in their own Group, Clone the group (then turn off the Group's visibility), Replicate the Clone. You will then have access to Play Frames, Random Start Frame, Source Start Frame and Source Frame Offset.

Play Frames will animate your replicated animations.

Random Start Frame will pick an offset into the animation and start the cell's animation from there.

Source Start Frame will allow you to keyframe the playback animation (Play Frames should be off) with all cells animating from the same frame.

Source Frame Offset will allow you to keyframe the playback all the cells using the Replicator's settings (order) for each cell. This means that if you have things like Shuffle Order set, then each cell animated will "proceed" as if Play Frames was selected, but under manual control...

I don't know if I can adequately explain this. You should just play around with it until you get a feel for it. A lot of different kinds of animations can be built this way.

Can you build what you want as an Emitter?