Inheritance is something which passes from the owner to the next owner, hisher heir. It could be in the form of material property or wealth, or it could be a spiritual inheritance such as succession in spiritual and non-material issues. When the inheritance is spiritual, it is more about mission and goals. If a prophet or any spiritual leader succeeds his predecessor, it mostly pertains to the mission and knowledge of the predecessor.
In the Holy Qur'an, we find the mention of inheritance of the Prophets (peace be upon them). Prophets have been bestowed heirs who inherited them and forwarded their mission and goals
A mention of the mercy of your Lord to His servant Zakariya. When he called upon his Lord in a low voice, He said My Lord! surely my bones are weakened and my head flares with hoariness, and, my Lord! I have never been unsuccessful in my prayer to Thee And surely I fear my cousins after me, and my wife is barren, therefore grant me from Thyself an heir, Who should inherit me and inherit from the children of Yaqub, and make him, my Lord, one in whom Thou art well pleased. O Zakariya! surely We give you good news of a boy whose name shall be Yahya We have not made before anyone his equal.
And Zakariya, when he cried to his Lord O my Lord leave me not alone; and Thou art the best of inheritors. So We responded to him and gave him Yahya and made his wife fit for him; surely they used to hasten, one with another, in deeds of goodness and to call upon Us, hoping and fearing and they were humble before Us.
And certainly We gave knowledge to Dawud and Sulayman, and they both said Praise be to Allah, Who has made us to excel many of His believing servants. And Sulayman was Dawud's heir, and he said O men! we have been taught the language of birds, and we have been given all things; most surely this is manifest grace.
And We gave to Dawud Sulayman, most excellent the servant! Surely he was frequent in returning (to Allah).
In the above verses, the Holy Qur'an talks about the Prophets and their heirs. The heirs of prophets can inherit them both materially and spiritually, but as the Prophets were more concerned about the continuation of their mission and Prophethood, it was more related to spiritual inheritance.
Likewise, the Qur'an also mentions the communities that inherited the divine Book and guidance from their predecessors
Then there came after them an evil posterity who inherited the Book, taking only the frail good of this low life and saying It will be forgiven us. And if the like good came to them, they would take it (too). Was not a promise taken from them in the Book that they would not speak anything about Allah but the truth, and they have read what is in it; and the abode of the hereafter is better for those who guard (against evil). Do you not then understand!?
Then We gave the Book for an inheritance to those whom We chose from among Our servants; but of them is he who makes his soul to suffer a loss, and of them is he who takes a middle course, and of them is he who is foremost in deeds of goodness by Allah's permission; this is the great excellence.  (3532)
And certainly We gave Musa the guidance, and We made the children of Israel inherit the Book.(4053)
The above verses dismiss the notion that inheritance of just guidance or book is sufficient. The verses lay emphasis over good deeds with faith and sincere adherence to the inherited teachings.
We often recite Ziyarat Waritha and acknowledge in it that Imam Hussain (peace be upon him) is the inheritor of the Prophets. We address the Imam in the following words

This inheritance which Imam Hussain has received from the Prophets is spiritual inheritance. He was the spiritual heir to the Prophets and he helped and saved their message. His blood which was spilled over the land of Karbala when he was suffering from thirst has been quenching the thirst of truthseekers, established an ever-illuminating line of demarcation between Truth and Falsehood, showed the precious value of inheritance of the Prophets, and set himself as an ever-abiding role model for all those who wish to follow and struggle for the Truth.
Now comes the question of who inherits the Great Message of Imam Hussain – who inherits Ashura
According to the rules of Kitab-ul-Miras (the Islamic law of inheritance), a person can be disqualified for inheritance if he is among the killers of the person to be inherited, and if he has forsaken his faith and adopted disbelief.
Having mentioned the above parameters for inheritance, we return to the question of inheritance of Ashura. Imam Hussain left a spiritual legacy, guidance, revolution and mission as his inheritance for us. His mission to save his grandfather's Ummah from deviation and rise against the persecutors is what he has left behind.
Now, for us to be his inheritors we are required to not join the party of Imam Hussain's killers. What were the motives of his killers These were to negate Allah's religion – as Yazid (may Allah remove His mercy from him) said that there was no revelation and there was no divine message – and to make the people return to the path of the accursed Shaytan. The Imam said
Indeed, these people (i.e. Yazid and the Umayyids) have committed themselves to the following of Satan and abandoned obedience to God. They have given currency to corruption, abolished the Islamic laws, plundered the public treasury, and made lawful what God has forbidden and unlawful what God has permitted. And I, of all people, have a greater right to act [in accordance with the Prophet's exhortation].
Indeed, I have not risen up to do mischief, neither as an adventurer, nor to cause corruption and tyranny. I have risen up solely to seek the reform of the Ummah of my grandfather. I want to enjoin what is good and stop what is wrong, and (in this) I follow the conduct of my grandfather and of my father Ali ibn Abi Talib.
If we negate the teachings of divine revelation, if we are arguing against the Islamic penal code or laws of inheritance. If we make fun of Hijab, if we argue that religion is not relevant in this age, if we do all this and still think that we are the heirs of Imam Hussain – we must keep in mind that by doing this, we are joining the party of his killers whose aims and goals were the same.
To be a true inheritor of the fruits of Ashura and the great revolution of Imam Hussain, we must be his true follower and uphold his goals, and should not join the party of his killers by abandoning Islam and trampling down his mission.