I believe it is true (Brits please correct me) that HMQ has certain rights over the persons of those who could succeed her.
This includes (at least for now) Archie and Lilibet, and it’s the reason Prince Markle Sparkle and the Mrs. cannot run the risk of transporting them to the UK.
Queen has no control over her succeeds her.
Succession is a matter of Parliamentary law, which I believe you know.
Not sure what you mean by ‘rights over the persons’. As far as the succession itself is concerned, she has none. All is determined by statute. As far as her more general rights are concerned, academic constitutional hypotheses aside, also effectively none other than those Parliament chooses to give her. In most respects, the monarch has less freedom of action than an ordinary citizen.