Is it permissible to pray over a deceased newborn after their burial, and is it permissible to exhume their grave to move them or change their burial place to a tomb?

May Allah magnify your reward and better your solace. We bring you glad tidings from what was narrated by Abu Musa al-Ash'ari, may Allah be pleased with him, that the Messenger of Allah, peace and blessings be upon him, said: "When a servant's child dies, Allah says to His angels: 'You have taken the child of My servant?' They say: 'Yes.' He says: 'You have taken the fruit of his heart?' They say: 'Yes.' He says: 'What did My servant say?' They say: 'He praised You and returned to You (i.e., said: 'Indeed, to Allah we belong, and to Him we shall return').' Allah says: 'Build for My servant a house in Paradise and call it the House of Praise (Bayt al-Hamd).'"

And from Mu'adh ibn Jabal, may Allah be pleased with him, from the Prophet, peace and blessings be upon him, who said: "By Him in Whose Hand is my soul, the miscarriage will drag its mother by its umbilical cord to Paradise if she seeks recompense for it."

The death of one's children and miscarriages are a shield from the Fire.

As for the prayer over a miscarriage (salat al-janazah (definition)), the Hanafis, Malikis, and Shafi'is are of the opinion that there is no obligation to pray over a miscarriage that died in its mother's womb, citing as evidence the hadith (definition): "If the miscarriage cries (upon birth), pray over it and it inherits."

However, Ahmad was of the opinion to pray over it based on the hadith: "And the miscarriage is prayed over, and forgiveness and mercy are supplicated for its parents," and because it is a soul into which the spirit has been breathed, so it is prayed over like one who cried out (upon birth).

Since the aforementioned child was buried according to a recognized school of thought, there is no need to dwell on the matter. You can pray over its grave, even if a month has passed since its burial. If you do not know the grave, you can pray over the entire cemetery, being in a state of ritual purity and fulfilling the rest of the conditions of prayer.