There’s a couple of ways this works:
The usual method is that Societies of Apostolic Life (of Pontifical Right) get invited into a diocese by a Bishop. They set up shop in the diocese, either by being assigned a parish (A), starting their own aposolate (B), or starting their own parish (C).
In the case of (A), the Bishop assigns them and lets them do their thing. The Society decides who will live and run the parish, and will handle the administrative aspects. The Society can decide to shuffle their resources around as they see fit. All forms of disciple still go through the Society. The Society can be removed from the parish by the Bishop, but not removed from the diocese. To be removed from the diocese, the bishop needs to go to the Congregation for Institutes of Consecrated Life and Societies of Apostolic Life.
In the case of (B), the Bishop cannot say anything regarding the aposolate if it is started and run by the Society. If the aposolate was started by the Diocese, the Bishop can shut it down, but still not remove the Society from the diocese.
In the case of (C), the Bishop has to go through the Congregation for Institutes of Consecrated Life and Societies of Apostolic Life for anything, really.
Societies of Apostolic Life that are Diocesan Right are totally under control of the Bishop, so if the group is this then ignore everything I said above. Once a group gets Pontifical Right, what I say above applies.
I’ve spent the last eleven years working with a Society of Apostolic Life, and was there when they received their Pontifical Right status, so I’m pretty familiar with how their systems work.