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Shiavault - a Vault of Shia Islamic Books Al-Muraja'at Letter 91 Rabi’ al-Awwal 19, 1330 I Justifying Their Behaviour Towards Usamah's Regiment, Yes, the Messenger of Allah (pbuh) had urged them to rush to participate in Usamah's campaign, ordering them to expedite, as you have mentioned, emphasizing his order till he told Usamah to invade Ubna in the morning, not allowing him to wait till the evening, telling him to go and emphasizing his order once more by telling him to rush.
But he, peace be upon him, according to all narrations, fell sick and started breathing very heavily, so much so that they started worrying about him and feeling too upset to leave him in such condition. They remained at Jurf waiting to know the condition of his health out of their own fear for his life and due to their attachment to him. They, therefore, are excused for having waited, and they should not be blamed.
As regarding their questioning the Prophet's appointment of Usamah after his death, in spite of what they remembered of his statements and his emphasis by word and by action, with which they were familiar, it was nothing more than their objection to his being too young, to his being a youth among middle-aged and old men. The latter naturally find it very difficult to receive orders from the young, and they by nature feel resentful towards submitting to their judgment.
Their hatred of his appointment was not an innovation but simply due to the human nature; so, consider that. As regarding their demand after the demise of the Messenger to depose Usamah, this is justified by some scholars among those who expected the Siddiq [Abu Bakr] to agree with them. These persons thought that such deposition would, in their view, serve the public interest.
Yet, for the sake of fairness, I personally cannot rationalize their request to depose him after seeing how angry the Prophet (pbuh) was when they requested the same, and his going out, feverish, bandaged and wrapped in a blanket, to denounce such a notion in his sermon from the pulpit. They knew that such an incident was, indeed, a historical milestone; so, their real motive is not known except by Allah.