They base their claims on saying that companions such as Sulayman b. Surad and Rifa'a b. Shaddad wrote to Imam Hussain (as) inviting him to Kufa to be their Imam, but then deserted him at the last minute. They also cite the Tawwabun uprising as "proof" that the Shi'a deserted Imam Hussain (as).
Thank you for your question. I think this kind of question arises from a misunderstanding of the historical circumstances of the people of Kufa and the success of Ibn Ziyad's policies in Kufa before the arrival of Imam Husayn (as). Not everyone in Kufa was a Shia and equating Kufans to Shias is not true or accurate. There were nobles in Kufa who stood to loose from the establishment of Imam Husayn (as) as the leader of Kufa and they helped Ibn Ziyad enforce tyrannical policies which induced an extreme amount of fear within the Kufan population such that all but the staunchest supporters of the Imam (as) chose silence and compliance over uprise.
For example, if any member of a tribe was seen to support the Imam, the whole tribe would be punished through the cutting off of stipends (which was the income for most people), not just the individual. This meant that before a person took action they had to think about all of their family members and their wider tribe including the women and children. They also had to bear in mind that many were willing to divulge any plans of disruption to Ibn Ziyad in order to protect their own skin and to seek favour with the government. It is in this way that many thousands of Kufans were forced on the battlefield but a much smaller percentage actually took part in the battle. People who wrote letters to the Imam (as) were not able to break out of these policies. It was also not certain for the people participating in the battle that the intention was to actually kill the Imam (as) until much later in the ordeal. I am certainly not justifying their lack of support of the Imam (as) at such a critical time, but am explaining the reason as to why these phenomena occurred and that is by coercion. Coercion does not prove anything other than the responsibility of the person who coerced, who was Ibn Ziyad.
The Tawwabun movement was also comprised of Shia that had been imprisoned and some historians have opined that it was the taking of the women as captives and their journey to Damascus that were significant causes for that movement. As for the blame for the death of the Imam (as), there is no doubt that it is on the immediate killers of the Imam (as) as well as all of those that were happy that he had been killed. There is no true Shia that would claim to be happy with the martyrdom of the Imam (as), whereas the tribe of Bani Umayyah celebrated the killing of the Imam (as) in the most flagrant manner.
May Allah's Mercy be distant from the killers of Imam Husayn (as) and may you always be successful.