How did the practice of matam or mourning for Hussain begin?

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How did the practice of matam or mourning for Hussain begin?
The first majlis-e-Hussain was recited in the market-place of Kufa by a lady (Hz Zainab binte Ali (AS)) from whose head her veil had been ripped off, whose hopes and aspirations had been destroyed on the blood-drenched sands of Kerbala but whose indomitable spirit stepped forward to free the Islamic values from the yoke of tyranny and oppression. 

"Woe upon you O people of Kufa. Do you realise which piece of Muhammad’s heart you have severed! Which pledge you have broken! Whose blood you have shed! Whose honour you have desecrated! It is not just Hussain whose headless body lies unburied on the sands of Kerbala. It is the heart of the Holy Prophet. It is the very soul of Islam!" 
Sermon of Janab Zainab Bint Ali in Market-place of Kufa 

When the prisoners were finally freed by Yezid, Bibi Zainab asked for an opportunity to have rites of remembrance in Damascus. A house was made available to them and aza-e-Hussain went on for over a week. Bibi Zainab (A.S.) laid the foundation of aza-e-Hussain in the very capital of his murderer! 

The first majlis touched and moved the people of Kufa so deeply as to give rise to both the Tawwabun movement and al-Mukhtar’s quest for vengeance. 

A year later when the shias were released from the prisons, and those who turned back due to fear, started to ponder over their actions of not helping their Imam. This lead to small pockets of rebellion against the Ummayads throughout the Islamic empire. 

One year after the event of Kerbala, 3000 faithful Shias started their movement as a revenge for Imam Hussain(AS). 

The martyrdom of Hussain was of great religious significance and had a deep heart-searching after-effect upon the Shi'is, giving a new turn to the mode and nature of the Shi'i movement. The tragic fate of the grandson of the Prophet stirred religious and moral sentiments, particularly among those of the Kufan followers of the House of the Prophet who had so zealously asked Hussain to come to Iraq to guide them on what they considered to be the path of God. But when Hussain came to Iraq they did not or could not stand with him in the hour of trial. Soon afterwards, however, they realized that their inability, or rather weakness, had been the cause of the tragedy. A deep sense of repentance set in, provoking their religious conscience; and in order to expiate their negligence and obtain God's forgiveness, they thought they must make similar sacrifices. They believed that they could only prove their real repentance by exposing themselves to death while seeking vengeance for the blood of Hussain. Hence they named themselves the Tawwabun (penitents) and are known in Islamic history by this self-imposed title.  

This movement, as will be seen below, proved to be an important step forward in the consolidation of Shi'i Islam. The movement began under the leadership of five of the oldest and most trusted associates of 'Ali, with a following of a hundred diehard and devoted Shi'is of Kufa, none of whom was below sixty years of age. This age factor should particularly be noted, as it indicates the maturity of their religious thinking and behaviour. The five leaders of the movement, Sulayman b. Surad al-Khuza`i, Al-Musayyab b. Najaba al-Fazari; `Abd Allah b. Sa`d b. Nufayl al-Azdi, Abd Allah b. Walin at-Tami, and Rifa`a b. Shaddad al-Bajali; had always been in the forefront of all Shi'i activities in Kufa, and were highly respected by the Shi'a for their sincerity of purpose and unshaken devotion to the cause of the Ahl al- Bayt. Similarly, the other hundred who joined these leaders of the movement are described as "the most select from among the followers of 'Ali". 
Tabari, II, p. 498; 
Baladhuri, V, pp.204 f. 

After the Tawabbun movement, 3 years after the event of Kerbala, Al-Mukhtar ibn Abi Ubayd al Thaqafi led another Shia rebellion against the Ummayads, more political in nature. Mukhtar however, was martyred a year later. 

This is also one of the reasons why we can see the graves of the Ahlulbayt, and the followers of Ali (AS), spread throughout Iran, Iraq, Saudia and Syria. This might be proof enough of the exile and execution, the Ahlulbayt and their Shias went through during the tyrant era of Ummayads and Abbasids.