Help on how to make a mechanical swing

I am trying to build a "Pokemon Go egg-hatching Lego swing thing".

I think I've nailed the "cradle" (that holds the phone) and also the stand which holds the cradle.

The thing I am strugling with is how to make a lego motor (Im using XL now) to swing the swing...

I want to build something that replicates this:

(the link above is marked so that it starts at 5mins where you get to see the "device")

Any ideas?

Edit: I managed to replicate this device pretty much 100%, but it seems this type of swing is not "compatible" with either my son or my own phone...so back to drawingboard I guess :D

Edit2: I posted something about this here: 

Edited August 6, 20222 yr by h3ctic
It failed :)