To what extent are the claims of people in Transoxiana that they are descendants of the Prophet, peace and blessings be upon him, valid? And should they be believed and treated accordingly, and what is their status in Islam?

The authenticity of a person's lineage to the Prophet (peace be upon him) cannot be definitively confirmed, and people are generally trusted regarding their lineages. If their lineage to the Ahl al-Bayt (family of the Prophet) is established, they are not permitted to receive zakat unless they have been denied their right to the khums (one-fifth of certain spoils/revenue), in which case they are allowed to take zakat as compensation. Their intermarriage with others is permissible, and there is no harm if they choose not to intermarry, as long as it does not lead to unjustly preventing their daughters from marrying. A Muslim should respect all Muslims, love them for the sake of Allah, communicate with them, and show kindness to them.