I find Exodus the most interesting one of these, I have an idea to add to it.
Diaspora promo could also allow Settler type units to move through and be expended in City-State territory if the city-state is at least Neutral. The city-state would gain population, though not as much as settling a new city would add (Settler: +1, Pioneer: +2, Colonist: +3). The city-state would also get a burst of Border Growth points (not sure how much, would increase with Pioneer and Colonist). The owner of the expended settler gains a small amount of influence and influence resting point with the City-State (+20 of both). It could also give the city-state free buildings that they didn't previously have if the unit was a Pioneer or Colonist... eeh, maybe not.
Diaspora settling in city-states would be inferior to sending diplomatic units in terms of getting large amounts of influence quick, but it would be useful for raising the influence resting point. Diaspora settling would also be useful for buffing city-states you already have a hold over. It should be balanced to be more of a niche thing to make use of the belief when there isn't good places to settle. It would also allow for settlers to pass through city-state territory without consequence of trespassing, which could come in handy if you need to get to a good spot fast and the quickest route would be through a city-state with its borders closed to you.