Is it a condition to remain in the same spot and not move or speak after the prayer in order to complete the prescribed remembrances (adhkar), so as to ensure the continued supplication of the angels for forgiveness, as has been mentioned?

The prescribed remembrance (dhikr) after prayer should be immediately following it or after a slight delay, so that the person is not considered to have turned away from the remembrance, or to have forgotten, or to have become preoccupied with something that came after the prayer, like Ayatul Kursi. The apparent meaning of the hadiths includes both obligatory (fard) and supererogatory (nafl) prayers, but most scholars have interpreted them as applying to obligatory prayers. A slight separation between the prayer and the prescribed remembrance, such as by speaking, does not affect it, just as departing from the place of prayer does not affect it. It is better for a Muslim to remain in their place of prayer until they have recited the remembrances, because the angels pray for them as long as they remain in their place of prayer.