What is the detailed explanation for moving the index finger during the Tashahhud, and is there a Hadith concerning that?

It is sunnah to point with the index finger and move it during the tashahhud. This is indicated by several hadiths, including:

The hadith of Wa'il ibn Hujr, in which the Prophet, peace and blessings be upon him, "raised his finger, and I saw him moving it, supplicating with it."

The hadith in Sahih Muslim from Ibn 'Umar, in which the Prophet, peace and blessings be upon him, "pointed with the finger next to the thumb." And in another narration by him, "He raised his right finger, the one next to the thumb, and supplicated with it."

The hadith of Sa'd, in which the Prophet, peace and blessings be upon him, said to him: "Make it one, make it one," and pointed with his index finger.

The hadith of Ibn 'Umar, in which the Prophet, peace and blessings be upon him, said: "It is harder on Satan than iron," meaning the index finger.

These hadiths indicate pointing with the index finger throughout the entire tashahhud and moving it during supplication. Sheikh Ibn 'Uthaymeen explained that the movement should accompany each supplicatory phrase in the tashahhud, indicating the loftiness of the One supplicated to, glorified be He.