Moving Finger in Tashahhud
Question

I've seen people move their finger up and down during tashahhud. Is this Sunnah?
Praise be to Allah.Is it sunnah to move the finger in tashahhud?
It is proven that the Prophet (peace and blessings of Allah be upon him) used to point with his index finger and move it during the tashahhud when praying. 
When to move finger in tashahhud
The scholars differed concerning that and there are several points of view. 

The Hanafis say that the finger should be raised when saying “La (no)” in the phrase “Ash-hadu an la ilaha ill-Allah (I bear witness that there is no god except Allah)” and it should be lowered when saying, “ill-Allah (except Allah).”  
The Shafi’is say that it should be raised when saying “ill-Allah.” 
The Malikis say that it should be moved right and left until one finishes the prayer. 
The Hanbalis say that one should point with the finger when saying the name of Allah, without moving it. 

Shaykh al-Albani (may Allah have mercy on him) said: “There is no basis for any of these definitions and manners in the Sunnah. The closest of them to the correct view is the Hanbali view, were it not that they limited raising the finger to when saying the name of Allah.” (Tamam al-Minnah, p. 223) 
Evidence for the permissibility of moving finger in tashahhud
With regard to the evidence concerning this issue: 

‘Abd-Allah ibn al-Zubayr said: When the Messenger of Allah (peace and blessings of Allah be upon him) sat during the prayer, he would place his left foot between his thigh and calf, and tuck his right foot underneath him, and place his left hand on his left knee, and place his right hand on his right thigh, and point with his finger. (Narrated by Muslim, 579)

In al-Nasai (1270) and Abu Dawud (989) it says: “He used to point with his finger when making du`a but he did not move it.” 
This addition – “but he did not move it” – was classed as da’if by Ibn al-Qayyim in Zad al-Ma’ad, 1/238. It was also classed as weak by al-Albani in Tamam al-Minnah, p. 218. 

Wail ibn Hajar said: I said: I will certainly watch how the Messenger of Allah (peace and blessings of Allah be upon him) prays. So I watched him and he stood up and said takbir (“Allahu akbar”), and raised his hands until they were level with his ears. Then he placed his right hand on his left hand, wrist and lower forearm. When he wanted to bow, he raised his hands likewise, and put his hands on his knees, and when he raised his head he raised his hands likewise. Then he prostrated and put his hands level with his ears, then he sat with his left foot tucked underneath him and put his left hand on his left thigh and knee, and he put the edge of his right elbow on his right thigh. Then he held two of his fingers and made a circle, then he raised his forefinger and moved it, making du`a with it. (Narrated by al-Nasai, 889; classed as sahih by Ibn Khuzaymah, 1/354; Ibn Majah, 5/170; classed as sahih by al-Albani in Irwa al-Ghalil, 367.) 

Shaykh Ibn ‘Uthaymin quoted this hadith – “moving it, making du`a with it” – as evidence that moving the forefinger during the tashahhud should be done with every phrase of the du`a. He said in al-Sharh al-Mumti’: 
“The Sunnah indicates that he should point with it when making du`a, because the wording of the hadith is “moving it, making du`a with it”. So every time you make du`a, move your finger thus indicating the exalted nature of the One to Whom you are addressing your du`as. So we say: 
“Al-salamu ‘alayka ayyuha’l-Nabiyyu (peace be upon you, O Prophet)” – you should point your finger because this salam is a kind of du`a. “Al-salamu ‘alayna (peace be upon us)” – you should point your finger. “Allahumma salli ‘ala Muhammad (O Allah, send blessings upon Muhammad)” – you should point your finger. “Allahumma barik ‘ala Muhammad) O Allah, send blessings upon Muhammad)” – you should point your finger. ‘A’idhu Billahi min ‘adhabi jahannam (I seek refuge with Allah from the torment of Hell)” – you should point your finger. “Wa min ‘adhab al-qabr (and from the torment of the grave)” – you should point your finger. “Wa min fitnat il-mahya wal-mamat (and from the trials of life and death)” – you should point your finger.  “Wa min fitnat il-masieh il-dajjal (and from the tribulation of the Dajjal)  – you should point your finger. Every time you make du`a you should point your finger, indicating the greatness of the One to Whom you are making du`a. This is closer to the Sunnah.” 
Looking at the finger when pointing
It is Sunnah when pointing to look at your finger. Al-Nawawi said: 
“The Sunnah is not to let your gaze go beyond the pointing finger. There is a sahih hadith concerning this in Sunan Abi Dawud. You should point in the direction of the qiblah and intend when pointing to affirm the Oneness of Allah and exclusive devotion to Him.” (Sharh Muslim, 5/81) 
The hadith to which al-Nawawi (may Allah have mercy on him) referred is the hadith of ‘Abd-Allah ibn al-Zubayr quoted above. The version narrated by Abu Dawud is (989): “And he should not let his gaze go beyond his pointing finger.” (Classed as sahih by al-Albani in Sahih Abi Dawud) 
Pointing with the finger towards the qiblah
It is Sunnah to point with it towards the qiblah. ‘Abd-Allah ibn ‘Umar narrated that he saw a man moving pebbles with his hand whilst he was praying. When he finished, ‘Abd-Allah said to him: “Do not move pebbles whilst you are praying, for that comes from the Shaytan. Rather do what the Messenger of Allah (peace and blessings of Allah be upon him) used to do. He put his right hand on his thigh and pointed with the finger that is next to the thumb towards the qiblah, and he fixed his gaze on it.” Then he said: “This is what I saw the Messenger of Allah (peace and blessings of Allah be upon him) doing.” (Narrated by al-Nasai, 1160; Ibn Khuzaymah, 1/355; Ibn Hibban, 5/273. classed as sahih by al-Albani in Sahih al-Nasai)
Bending the finger slightly when pointing 
Bending the finger slightly when pointing was mentioned in the hadith of Numayr al-Khuza’i, (Narrated by Abu Dawud, 991, and al-Nasai, 1275) But this is a weak hadith. (See Tamam al-Minnah by al-Albani p. 222)
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And Allah knows best.