Recreate this in Tyflow?

[rusty]

I initially thought it was some sort of noise or displacement but the motion has a bit of falloff, delay to it, almost wave like.

You can make falloff, by using speed operator, that will push particles in one frame up, then use force operator with gravity applied, that will then slowly take over, and make them fall. 

I did a quick example on how I would approach this, and will try explain what I did for anyone interested. 

First I created a plane 15x15 pixels. 
Then I used birth surface so I have one particle per each vertex. 
I then use send out operator so I send all particles in new event, simply because I wanted to use loop, and I can not connect it to the event with birth operator, only event without birth operator. 
I then applied noise material to the original plane, and animated phase on it. 
I used surface test referencing original plane, and I copy - instanced noise map to that event. so that way it will trigger particles to go to new event. 
In the new event I did what I wrote initially.. just a speed and gravity... since the particles would fall to infinity, I added property test where I told particles if they go bellow zero in Z exis, go to new event to just stop them. You could also use collision operator for this, with zero bounce and 100 friction (if you have uneven surface). 
When they stop in the last event, I just simply copy-instanced same surface test as in the second event, and connected it to second event (I guess, send out would also work here). 
Finally I added shape operator so I make little spheres for the particles.
And I added spline paths (by neighbours) so it connect close particles with splines. 
If you grab TySpline object in tyflow, and go to tyspline modifier, under neighbor splines you can adjust distance to your liking... and in the tyspline mesher you can adjust how fat you want that spline to be. 

Attaching sample file.

Nice! Thanks for sharing!