If Hashem responds positively to the Rebbe or great Rav after others have davened and not seen a response, surely this does not mean that Hashem changes His mind, so as to speak, when the Rebbe davens. How does it work?

Each and every prayer is heard and taken into account, we do not know which prayer is the one that Hashem responded to, it may be that another prayer, and another one made the difference at the end. Rabbi Ro'i Margalit