I found this document, excerpted here, that outlines the origins of commemoration of Ashoura - specifically with majlis and recitations - other practices that are observed by some are not mentioned here.
The first majlis-e-Hussain was recited in the market-place of Kufa by a lady 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. Standing on her unsaddled camel, she looked at the multitude rejoicing the victory of Yezid. As soon as people saw her, they were quiet. They knew that a historic moment for Kufa had arrived. Looking straight at them, the daughter of Ali said:
"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!"
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.
When the news of tragedy reached Medina in the third week of Muharram there was such intense weeping and wailing from the homes of Banu Hashim that the very walls of masjidun-nabawi began to tremble. Zainab, Umme Luqman, the daughter of Aqeel ibne Abi Talib came out screaming: "What will you say when the Prophet asks you: "What have you, the last ummah, done with my offspring and my family after I left them? Some of them are prisoners and some of them lie killed, stained with blood. What sort of ajr-e-risaalah is this that you disobey me by oppressing my children ?"
Fatimah Binte Huzaam, also known as Ummul Baneen, carried her young grandson Ubaidullah ibne Abbas and prepared to go out. When asked where she was going, she said that she was taking the orphan of Abbas to offer condolences to the mother of Hussain.
Marwan ibne Hakam reports that every afternoon men and women would gather at Jannat-ul-Baqee and there would be remembrance of the tragedy of Kerbala and the weeping and wailing could be heard miles away.
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!
On their return to Madina, Bibi Zainab (A.S.) took over the leadership of aza-e-Hussain in the city of the Holy Prophet. This aroused such strong emotions in the people and such revulsion against the oppressor that Amr ibne Said ibne al-Aas wrote to Yezid to have Bibi Zainab exiled from Madina. This was done in the beginning of 62 A.H. Bibi Zainab (A.S.) died shortly afterwards.
We have no record of public orations by our Imams about the tragedy of Kerbala. We have, however, several ahadeeth about the merits of participating in the mourning ceremonies. In this connection we must remember that the regime was hostile to the shiahs and was anxious to cover up the tragedy of Kerbala.
Imam Muhammad Baqir (A.S.) issued a directive which gave a definite form to the keeping of the memory of Imam Hussain (A.S.) alive. He recommended that for those believers for whom it was possible and convenient they should go for the ziyarah of the grave of Imam Hussain. For those for whom it was not possible or convenient, they should gather together and hold mourning ceremony and weep. Ibn Qawlawayah p. 104
There is also the following tradition reported from the fifth Imam:
May Allah have mercy on a man who meets with another in order to remember our situation. There will be an angel with them who will seek forgiveness for them…………..If you gather together and occupy yourselves in remembering us, then our memory will be kept alive in your meetings and remembrances. The best of people after us are those who remember our situation and urge others to remember us. Ibn Qawlawayah p. 174/5
It is reported that al-Fudhayl Ibne Yasaar came to pay his respects to the Imam Ja’far Sadiq (A.S.)
After the exchange of usual courtesies, Imam asked al-Fudhayl: "Do you people ever organise majaalis to recall the martyrdom of Imam Hussain?" Al-Fudhayl, with tears pouring down his eyes, replied: "Yabna Rasulillah, indeed we do." The Imam said: "May Allah bless you. I highly approve of such majaalis."
It must be borne in mind that the Arabs mostly expressed their emotion through poetry. Poetry thus became the medium of describing the horrors of the tragedy of Kerbala, the cause of Imam Hussain and the atrocities which the ahlul-bayt were made to endure. There are today extant several poems which the poets recited in presence of our holy Imams and as such can be regarded as having been approved by them both as to form and substance.
The only historical account in prose that was written not long after the massacre of Kerbala was that of Abi Mikhnaf. His account is relied upon both by Tabari and Shaykh Mufeed (A.R.). Many other accounts were written and published after the ghaybah. The most well known amongst these are the Aamali by Shaykh Suduq (A.R.) and the great work of Allamah Majlisi (A.R.), the Bihar-ul-Anwaar.
While we have evidence of many eminent fuqaha and muhadditheen lecturing to their students on the various aspects of Kerbala, we can not assert with any confidence that they delivered public lectures on the subject. It is, however, authoritatively reported that Shaykh Allamah Majlisi and Shaykh Shushtari, whenever they spoke, whether to the students or in the public, they would end their lecture with a brief reference to the masa’ib of Imam Hussain.
by Muslim Bhanji