Gradual transition between keyed particle objects?

I know how to do this, where the particles arrive at their destination all at once:

But not this, where the particle arrive gradually:

Sometimes you can change the interpretation by juggling the end frame on the control objects particle system which will extend the particles out and playing with the random. Too bad you can’t keyframe the end frame on that object to control it by time.

Found that out. The problem now is that I’m keying from a particles system with Newtonian physics, but that needs to be instantly hidden, and now the keyed particles spawn in gradually meaning the transition is not smooth.

Might not be possible. Unless I convert the physics simulation to keyframes, then physics could be turned off?